Axiomatic Approach to probability and Properties of Probability Measure Economics Notes | EduRev

Economics : Axiomatic Approach to probability and Properties of Probability Measure Economics Notes | EduRev

 Page 1


 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   1 
 
MODULE 1 
PROBABILITY  
LECTURE 2 
Topics 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
 
In the following section we will discuss the modern approach to probability theory where 
we will not be concerned with how probabilities are assigned to suitably chosen subsets 
of ?? . Rather we will define the concept of probability for certain types of subsets ?? using 
a set of axioms that are consistent with properties (i)-(iii) of classical (or relative 
frequency) method. We will also study various properties of probability measures.  
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
We begin this section with the following definitions. 
Definition 2.1 
(i)  A set whose elements are themselves set is called a class of sets. A class of sets 
will be usually denoted by script letters ?? ,B,?? ,… . For example ?? =
  1 , 1, 3 , 2, 5, 6  ; 
(ii) Let ?? be a class of sets. A function ?? :?? ?R is called a set function. In other 
words, a real-valued function whose domain is a class of sets is called a set 
function. _ 
As stated above, in many situations, it may not be possible to assign probabilities to all 
subsets of the sample space ?? such that properties (i)-(iii) of classical (or relative 
frequency) method are satisfied. Therefore one begins with assigning probabilities to 
members of an appropriately chosen class ?? of subsets of ?? (e.g., if ?? =R, then ?? may 
be class of all open intervals in R; if ?? is a countable  set, then ?? may be class of all 
singletons  ?? ,?? ??? ). We call the members of ?? as basic sets. Starting from the basic 
sets in ?? assignment of probabilities is extended, in an intuitively justified manner, to as 
many subsets of ?? as possible keeping in mind that properties (i)-(iii) of classical (or 
Page 2


 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   1 
 
MODULE 1 
PROBABILITY  
LECTURE 2 
Topics 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
 
In the following section we will discuss the modern approach to probability theory where 
we will not be concerned with how probabilities are assigned to suitably chosen subsets 
of ?? . Rather we will define the concept of probability for certain types of subsets ?? using 
a set of axioms that are consistent with properties (i)-(iii) of classical (or relative 
frequency) method. We will also study various properties of probability measures.  
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
We begin this section with the following definitions. 
Definition 2.1 
(i)  A set whose elements are themselves set is called a class of sets. A class of sets 
will be usually denoted by script letters ?? ,B,?? ,… . For example ?? =
  1 , 1, 3 , 2, 5, 6  ; 
(ii) Let ?? be a class of sets. A function ?? :?? ?R is called a set function. In other 
words, a real-valued function whose domain is a class of sets is called a set 
function. _ 
As stated above, in many situations, it may not be possible to assign probabilities to all 
subsets of the sample space ?? such that properties (i)-(iii) of classical (or relative 
frequency) method are satisfied. Therefore one begins with assigning probabilities to 
members of an appropriately chosen class ?? of subsets of ?? (e.g., if ?? =R, then ?? may 
be class of all open intervals in R; if ?? is a countable  set, then ?? may be class of all 
singletons  ?? ,?? ??? ). We call the members of ?? as basic sets. Starting from the basic 
sets in ?? assignment of probabilities is extended, in an intuitively justified manner, to as 
many subsets of ?? as possible keeping in mind that properties (i)-(iii) of classical (or 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   2 
 
relative frequency) method are not violated. Let us denote by F the class of sets for 
which the probability assignments can be finally done. We call the class F as event space 
and elements of F are called events. It will be reasonable to assume that F satisfies the 
following properties: (i) ?? ?F, (ii)?? ?F??? ?? =?? -?? ?F , and (iii)?? ?? ?F,?? =
1,2,…? ?? ?? ?F
8
?? =1
. This leads to introduction of the following definition. 
Definition 2.2 
 A sigma-field (?? -field) of subsets of ?? is a class F of subsets of ?? satisfying the 
following properties: 
(i) ?? ?F; 
(ii) ?? ?F??? ?? =?? -?? ?F  (closed under complements); 
(iii) ?? ?? ?F,?? = 1, 2,…? ?? ?? ?F
8
?? =1
 (closed under countably infinite unions). _ 
Remark 2.1 
(i) We expect the event space to be a ?? -field; 
(ii)  Suppose that F is a ?? -field of subsets of ?? . Then, 
(a) ?? ?F since  ?? =?? ??  
(b) ?? 1
,?? 2
,…?F? ?? ?? ?F
8
?? =1
 since ?? ?? 8
?? =1
=  ?? ?? ?? 8
?? =1
 
?? ; 
(c) ?? ,?? ?F??? -?? =?? n?? ?? ?F and ?? ??? ? ?? -?? ? ?? -?? ?F; 
(d) ?? 1
,?? 2
,… ,?? ?? ?F, for some ?? ?N,? ?? ?? ?F
?? ?? =1
 and  ?? ?? ?F
?? ?? =1
 (take 
?? ?? +1
=?? ?? +2
=? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 or ?? ?? +1
=?? ?? +2
=
? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 ); 
(e) although the power set of ??  ?? ??   is a ?? -field of subsets of ?? , in 
general, a ?? -field may not contain all subsets of ?? . _ 
Example 2.1 
(i) F = ?? ,??  is a sigma field, called the trivial sigma-field; 
(ii) Suppose that ?? ??? . Then F = ?? ,?? ?? ,?? ,??  is a ?? -field of subsets of ?? . It is 
the smallest sigma-field containing the set ?? ; 
(iii) Arbitrary intersection of ?? -fields is a ?? -field (see Problem 3 (i));  
(iv) Let ?? be a class of subsets of ?? and let  ?? ?? :?? ???  be the collection of all ?? -
fields that contain ?? . Then 
                                                F = F
?? ?? ??? 
is a ?? -field and it is the smallest ?? -field that contains class ?? (called the ?? -
field generated by ?? and is denoted by ?? (?? )) (see Problem 3 (iii)); 
(v) Let ?? =R and let ?? be the class of all open intervals in R. Then B
1
=?? ??  is 
called the Borel ?? -field on R. The Borel ?? -field in R
?? (denoted by B
?? ) is the 
Page 3


 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   1 
 
MODULE 1 
PROBABILITY  
LECTURE 2 
Topics 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
 
In the following section we will discuss the modern approach to probability theory where 
we will not be concerned with how probabilities are assigned to suitably chosen subsets 
of ?? . Rather we will define the concept of probability for certain types of subsets ?? using 
a set of axioms that are consistent with properties (i)-(iii) of classical (or relative 
frequency) method. We will also study various properties of probability measures.  
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
We begin this section with the following definitions. 
Definition 2.1 
(i)  A set whose elements are themselves set is called a class of sets. A class of sets 
will be usually denoted by script letters ?? ,B,?? ,… . For example ?? =
  1 , 1, 3 , 2, 5, 6  ; 
(ii) Let ?? be a class of sets. A function ?? :?? ?R is called a set function. In other 
words, a real-valued function whose domain is a class of sets is called a set 
function. _ 
As stated above, in many situations, it may not be possible to assign probabilities to all 
subsets of the sample space ?? such that properties (i)-(iii) of classical (or relative 
frequency) method are satisfied. Therefore one begins with assigning probabilities to 
members of an appropriately chosen class ?? of subsets of ?? (e.g., if ?? =R, then ?? may 
be class of all open intervals in R; if ?? is a countable  set, then ?? may be class of all 
singletons  ?? ,?? ??? ). We call the members of ?? as basic sets. Starting from the basic 
sets in ?? assignment of probabilities is extended, in an intuitively justified manner, to as 
many subsets of ?? as possible keeping in mind that properties (i)-(iii) of classical (or 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   2 
 
relative frequency) method are not violated. Let us denote by F the class of sets for 
which the probability assignments can be finally done. We call the class F as event space 
and elements of F are called events. It will be reasonable to assume that F satisfies the 
following properties: (i) ?? ?F, (ii)?? ?F??? ?? =?? -?? ?F , and (iii)?? ?? ?F,?? =
1,2,…? ?? ?? ?F
8
?? =1
. This leads to introduction of the following definition. 
Definition 2.2 
 A sigma-field (?? -field) of subsets of ?? is a class F of subsets of ?? satisfying the 
following properties: 
(i) ?? ?F; 
(ii) ?? ?F??? ?? =?? -?? ?F  (closed under complements); 
(iii) ?? ?? ?F,?? = 1, 2,…? ?? ?? ?F
8
?? =1
 (closed under countably infinite unions). _ 
Remark 2.1 
(i) We expect the event space to be a ?? -field; 
(ii)  Suppose that F is a ?? -field of subsets of ?? . Then, 
(a) ?? ?F since  ?? =?? ??  
(b) ?? 1
,?? 2
,…?F? ?? ?? ?F
8
?? =1
 since ?? ?? 8
?? =1
=  ?? ?? ?? 8
?? =1
 
?? ; 
(c) ?? ,?? ?F??? -?? =?? n?? ?? ?F and ?? ??? ? ?? -?? ? ?? -?? ?F; 
(d) ?? 1
,?? 2
,… ,?? ?? ?F, for some ?? ?N,? ?? ?? ?F
?? ?? =1
 and  ?? ?? ?F
?? ?? =1
 (take 
?? ?? +1
=?? ?? +2
=? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 or ?? ?? +1
=?? ?? +2
=
? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 ); 
(e) although the power set of ??  ?? ??   is a ?? -field of subsets of ?? , in 
general, a ?? -field may not contain all subsets of ?? . _ 
Example 2.1 
(i) F = ?? ,??  is a sigma field, called the trivial sigma-field; 
(ii) Suppose that ?? ??? . Then F = ?? ,?? ?? ,?? ,??  is a ?? -field of subsets of ?? . It is 
the smallest sigma-field containing the set ?? ; 
(iii) Arbitrary intersection of ?? -fields is a ?? -field (see Problem 3 (i));  
(iv) Let ?? be a class of subsets of ?? and let  ?? ?? :?? ???  be the collection of all ?? -
fields that contain ?? . Then 
                                                F = F
?? ?? ??? 
is a ?? -field and it is the smallest ?? -field that contains class ?? (called the ?? -
field generated by ?? and is denoted by ?? (?? )) (see Problem 3 (iii)); 
(v) Let ?? =R and let ?? be the class of all open intervals in R. Then B
1
=?? ??  is 
called the Borel ?? -field on R. The Borel ?? -field in R
?? (denoted by B
?? ) is the 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   3 
 
?? -field  generated by class of all open rectangles in R
?? . A set ?? ?B
?? is called 
a Borel set in R
?? ; here R
?? = {(?? 1
,… ,?? ?? ):-8 <?? ?? < 8, ?? = 1,… ,?? }  
denotes the ?? -dimensional Euclidean space; 
(vi) B
1
 contains all singletons and hence all countable subsets of  
 R  ?? =  ?? -
1
?? ,?? +
1
?? 
8
?? =1
  . _ 
Let ?? be an appropriately chosen class of basic subsets of ?? for which the probabilities 
can be assigned to begin with (e.g., if ?? =R then ?? may be class of all open intervals in 
R; if ?? is a countable set then ?? may be class of all singletons  ?? ,?? ??? ). It turns out (a 
topic for an advanced course in probability theory) that, for an appropriately chosen class 
?? of basic sets, the assignment of probabilities that is consistent with properties (i)-(iii) of 
classical (or relative frequency) method can be extended in an unique manner from ?? to 
?? (?? ), the smallest ?? -field containing the class ?? . Therefore, generally the domain F of a 
probability measure is taken to be ?? (?? ), the ?? -field generated by the class ?? of basic 
subsets of ?? .  We have stated before that we will not care about how assignment of 
probabilities to various members of event space F (a ?? -field of subsets of ?? ) is done. 
Rather we will be interested in properties of probability measure defined on event space 
F. 
Let ?? be a sample space associated with a random experiment and let F be the event 
space (a ?? -field of subsets of ?? ). Recall that members of F are called events. Now we 
provide a mathematical definition of probability based on a set of axioms. 
Definition 2.3 
(i) Let F be a ?? -field of subsets of ?? . A probability function (or a probability 
measure) is a set function ?? , defined on F, satisfying the following three axioms: 
 
(a) ?? ?? = 0,   ??? ?F;                         (Axiom 1: Non- negativity);                         
(b) If ?? 1
,?? 2
,… is a countably infinite collection of mutually exclusive events 
 i. e., ?? ?? ?F, ?? = 1, 2,… ,?? ?? n?? ?? =?? ,?? ???  then 
??  ?? ?? 8
?? =1
 = ?? ?? ?? 
8
1=1
;               (Axiom 2: Countably infinite additive)  
(c) ?? ?? = 1             (Axiom 3: Probability of the sample space is 1). 
 
(ii) The triplet  ?? ,F,??  is called a probability space. _ 
 
 
Page 4


 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   1 
 
MODULE 1 
PROBABILITY  
LECTURE 2 
Topics 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
 
In the following section we will discuss the modern approach to probability theory where 
we will not be concerned with how probabilities are assigned to suitably chosen subsets 
of ?? . Rather we will define the concept of probability for certain types of subsets ?? using 
a set of axioms that are consistent with properties (i)-(iii) of classical (or relative 
frequency) method. We will also study various properties of probability measures.  
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
We begin this section with the following definitions. 
Definition 2.1 
(i)  A set whose elements are themselves set is called a class of sets. A class of sets 
will be usually denoted by script letters ?? ,B,?? ,… . For example ?? =
  1 , 1, 3 , 2, 5, 6  ; 
(ii) Let ?? be a class of sets. A function ?? :?? ?R is called a set function. In other 
words, a real-valued function whose domain is a class of sets is called a set 
function. _ 
As stated above, in many situations, it may not be possible to assign probabilities to all 
subsets of the sample space ?? such that properties (i)-(iii) of classical (or relative 
frequency) method are satisfied. Therefore one begins with assigning probabilities to 
members of an appropriately chosen class ?? of subsets of ?? (e.g., if ?? =R, then ?? may 
be class of all open intervals in R; if ?? is a countable  set, then ?? may be class of all 
singletons  ?? ,?? ??? ). We call the members of ?? as basic sets. Starting from the basic 
sets in ?? assignment of probabilities is extended, in an intuitively justified manner, to as 
many subsets of ?? as possible keeping in mind that properties (i)-(iii) of classical (or 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   2 
 
relative frequency) method are not violated. Let us denote by F the class of sets for 
which the probability assignments can be finally done. We call the class F as event space 
and elements of F are called events. It will be reasonable to assume that F satisfies the 
following properties: (i) ?? ?F, (ii)?? ?F??? ?? =?? -?? ?F , and (iii)?? ?? ?F,?? =
1,2,…? ?? ?? ?F
8
?? =1
. This leads to introduction of the following definition. 
Definition 2.2 
 A sigma-field (?? -field) of subsets of ?? is a class F of subsets of ?? satisfying the 
following properties: 
(i) ?? ?F; 
(ii) ?? ?F??? ?? =?? -?? ?F  (closed under complements); 
(iii) ?? ?? ?F,?? = 1, 2,…? ?? ?? ?F
8
?? =1
 (closed under countably infinite unions). _ 
Remark 2.1 
(i) We expect the event space to be a ?? -field; 
(ii)  Suppose that F is a ?? -field of subsets of ?? . Then, 
(a) ?? ?F since  ?? =?? ??  
(b) ?? 1
,?? 2
,…?F? ?? ?? ?F
8
?? =1
 since ?? ?? 8
?? =1
=  ?? ?? ?? 8
?? =1
 
?? ; 
(c) ?? ,?? ?F??? -?? =?? n?? ?? ?F and ?? ??? ? ?? -?? ? ?? -?? ?F; 
(d) ?? 1
,?? 2
,… ,?? ?? ?F, for some ?? ?N,? ?? ?? ?F
?? ?? =1
 and  ?? ?? ?F
?? ?? =1
 (take 
?? ?? +1
=?? ?? +2
=? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 or ?? ?? +1
=?? ?? +2
=
? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 ); 
(e) although the power set of ??  ?? ??   is a ?? -field of subsets of ?? , in 
general, a ?? -field may not contain all subsets of ?? . _ 
Example 2.1 
(i) F = ?? ,??  is a sigma field, called the trivial sigma-field; 
(ii) Suppose that ?? ??? . Then F = ?? ,?? ?? ,?? ,??  is a ?? -field of subsets of ?? . It is 
the smallest sigma-field containing the set ?? ; 
(iii) Arbitrary intersection of ?? -fields is a ?? -field (see Problem 3 (i));  
(iv) Let ?? be a class of subsets of ?? and let  ?? ?? :?? ???  be the collection of all ?? -
fields that contain ?? . Then 
                                                F = F
?? ?? ??? 
is a ?? -field and it is the smallest ?? -field that contains class ?? (called the ?? -
field generated by ?? and is denoted by ?? (?? )) (see Problem 3 (iii)); 
(v) Let ?? =R and let ?? be the class of all open intervals in R. Then B
1
=?? ??  is 
called the Borel ?? -field on R. The Borel ?? -field in R
?? (denoted by B
?? ) is the 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   3 
 
?? -field  generated by class of all open rectangles in R
?? . A set ?? ?B
?? is called 
a Borel set in R
?? ; here R
?? = {(?? 1
,… ,?? ?? ):-8 <?? ?? < 8, ?? = 1,… ,?? }  
denotes the ?? -dimensional Euclidean space; 
(vi) B
1
 contains all singletons and hence all countable subsets of  
 R  ?? =  ?? -
1
?? ,?? +
1
?? 
8
?? =1
  . _ 
Let ?? be an appropriately chosen class of basic subsets of ?? for which the probabilities 
can be assigned to begin with (e.g., if ?? =R then ?? may be class of all open intervals in 
R; if ?? is a countable set then ?? may be class of all singletons  ?? ,?? ??? ). It turns out (a 
topic for an advanced course in probability theory) that, for an appropriately chosen class 
?? of basic sets, the assignment of probabilities that is consistent with properties (i)-(iii) of 
classical (or relative frequency) method can be extended in an unique manner from ?? to 
?? (?? ), the smallest ?? -field containing the class ?? . Therefore, generally the domain F of a 
probability measure is taken to be ?? (?? ), the ?? -field generated by the class ?? of basic 
subsets of ?? .  We have stated before that we will not care about how assignment of 
probabilities to various members of event space F (a ?? -field of subsets of ?? ) is done. 
Rather we will be interested in properties of probability measure defined on event space 
F. 
Let ?? be a sample space associated with a random experiment and let F be the event 
space (a ?? -field of subsets of ?? ). Recall that members of F are called events. Now we 
provide a mathematical definition of probability based on a set of axioms. 
Definition 2.3 
(i) Let F be a ?? -field of subsets of ?? . A probability function (or a probability 
measure) is a set function ?? , defined on F, satisfying the following three axioms: 
 
(a) ?? ?? = 0,   ??? ?F;                         (Axiom 1: Non- negativity);                         
(b) If ?? 1
,?? 2
,… is a countably infinite collection of mutually exclusive events 
 i. e., ?? ?? ?F, ?? = 1, 2,… ,?? ?? n?? ?? =?? ,?? ???  then 
??  ?? ?? 8
?? =1
 = ?? ?? ?? 
8
1=1
;               (Axiom 2: Countably infinite additive)  
(c) ?? ?? = 1             (Axiom 3: Probability of the sample space is 1). 
 
(ii) The triplet  ?? ,F,??  is called a probability space. _ 
 
 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   4 
 
Remark 2.2 
(i) Note that if ?? 1
,?? 2
,… is a countably infinite collection of sets in a ?? -field F then 
 ?? ?? 8
?? =1
?F and, therefore, ?? ( ?? ?? 8
?? =1
) is well defined; 
(ii) In any probability space  ?? ,F,??  we have ?? ?? = 1 (or ?? ?? = 0; see Theorem 
2.1 (i) proved later) but if ?? ?? = 1 (or ?? ?? = 0), for some ?? ?F, then it does 
not mean that ?? =?? ( or ?? =?? ) (see Problem 14 (ii)). 
(iii) In general not all subsets of ?? are events, i.e., not all subsets of ?? are elements of 
F. 
(iv)  When ?? is countable it is possible to assign probabilities to all subsets of ?? using 
Axiom 2 provided we can assign probabilities to singleton subsets  ??  of ?? . To 
illustrate this let ?? = ?? 1
,?? 2
,…   or ?? = ?? 1
,… ,?? ?? , for some n ?N  and let 
??  ?? ??  =?? ?? , ?? = 1, 2,…, so that 0=?? ?? = 1, ?? = 1,2,… (see Theorem 2.1 (iii) 
below) and  ?? ?? =
8
?? =1
 ??  ?? ??  
8
?? =1
=??   ?? ?? 
8
?? =1
 =?? ?? = 1. Then, for any 
?? ??? , 
?? ?? =  ?? ?? .
?? :?? ?? ??? 
Thus in this case we may take F =?? ?? , the power set of ?? . It is worth 
mentioning here that if ?? is countable and ?? =  ?? : ?? ???  (class of all 
singleton subsets of ?? ) is the class of basic sets for which the assignment of the 
probabilities can be done, to begin with, then ?? (?? ) =?? ??  (see Problem 5 (ii)).  
(v) Due to some inconsistency problems, assignment of probabilities for all subsets of 
?? is not possible when ?? is continuum (e.g., if ?? contains an interval). _ 
Theorem 2.1 
Let  ?? ,F,??  be a probability space. Then 
(i) ?? ?? = 0; 
(ii) ?? ?? ?F,?? = 1, 2,… .?? , and ?? ?? n?? ?? =?? , ?? ??? ???  ?? ?? ?? ?? =1
 = ?? ?? ?? 
?? ?? =1
 
(finite additivity); 
(iii) ??? ?F, 0=?? ?? = 1 and ?? ?? ?? = 1-?? ?? ; 
(iv) ?? 1
,?? 2
?F and?? 1
??? 2
??? ?? 2
-?? 1
 =?? ?? 2
 -?? ?? 1
 and?? ?? 1
 =?? ?? 2
  
(monotonicity of probability measures); 
(v) ?? 1
,?? 2
?F??? ?? 1
??? 2
 =?? ?? 1
 +?? ?? 2
 -?? ?? 1
n?? 2
 . 
Proof. 
(i) Let ?? 1
=?? and ?? ?? =?? , ?? = 2, 3,…. Then ?? ?? 1
 = 1, (Axiom 3), ?? ?? ?F, ?? =
1, 2,… , ?? 1
= ?? ?? 8
?? =1
 and ?? ?? n?? ?? =?? , ?? ??? . Therefore,  
Page 5


 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   1 
 
MODULE 1 
PROBABILITY  
LECTURE 2 
Topics 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
 
In the following section we will discuss the modern approach to probability theory where 
we will not be concerned with how probabilities are assigned to suitably chosen subsets 
of ?? . Rather we will define the concept of probability for certain types of subsets ?? using 
a set of axioms that are consistent with properties (i)-(iii) of classical (or relative 
frequency) method. We will also study various properties of probability measures.  
1.2 AXIOMATIC APPROACH TO PROBABILITY AND 
PROPERTIES OF PROBABILITY MEASURE 
We begin this section with the following definitions. 
Definition 2.1 
(i)  A set whose elements are themselves set is called a class of sets. A class of sets 
will be usually denoted by script letters ?? ,B,?? ,… . For example ?? =
  1 , 1, 3 , 2, 5, 6  ; 
(ii) Let ?? be a class of sets. A function ?? :?? ?R is called a set function. In other 
words, a real-valued function whose domain is a class of sets is called a set 
function. _ 
As stated above, in many situations, it may not be possible to assign probabilities to all 
subsets of the sample space ?? such that properties (i)-(iii) of classical (or relative 
frequency) method are satisfied. Therefore one begins with assigning probabilities to 
members of an appropriately chosen class ?? of subsets of ?? (e.g., if ?? =R, then ?? may 
be class of all open intervals in R; if ?? is a countable  set, then ?? may be class of all 
singletons  ?? ,?? ??? ). We call the members of ?? as basic sets. Starting from the basic 
sets in ?? assignment of probabilities is extended, in an intuitively justified manner, to as 
many subsets of ?? as possible keeping in mind that properties (i)-(iii) of classical (or 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   2 
 
relative frequency) method are not violated. Let us denote by F the class of sets for 
which the probability assignments can be finally done. We call the class F as event space 
and elements of F are called events. It will be reasonable to assume that F satisfies the 
following properties: (i) ?? ?F, (ii)?? ?F??? ?? =?? -?? ?F , and (iii)?? ?? ?F,?? =
1,2,…? ?? ?? ?F
8
?? =1
. This leads to introduction of the following definition. 
Definition 2.2 
 A sigma-field (?? -field) of subsets of ?? is a class F of subsets of ?? satisfying the 
following properties: 
(i) ?? ?F; 
(ii) ?? ?F??? ?? =?? -?? ?F  (closed under complements); 
(iii) ?? ?? ?F,?? = 1, 2,…? ?? ?? ?F
8
?? =1
 (closed under countably infinite unions). _ 
Remark 2.1 
(i) We expect the event space to be a ?? -field; 
(ii)  Suppose that F is a ?? -field of subsets of ?? . Then, 
(a) ?? ?F since  ?? =?? ??  
(b) ?? 1
,?? 2
,…?F? ?? ?? ?F
8
?? =1
 since ?? ?? 8
?? =1
=  ?? ?? ?? 8
?? =1
 
?? ; 
(c) ?? ,?? ?F??? -?? =?? n?? ?? ?F and ?? ??? ? ?? -?? ? ?? -?? ?F; 
(d) ?? 1
,?? 2
,… ,?? ?? ?F, for some ?? ?N,? ?? ?? ?F
?? ?? =1
 and  ?? ?? ?F
?? ?? =1
 (take 
?? ?? +1
=?? ?? +2
=? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 or ?? ?? +1
=?? ?? +2
=
? =?? so that  ?? ?? ?? ?? =1
= ?? ?? 8
?? =1
 ); 
(e) although the power set of ??  ?? ??   is a ?? -field of subsets of ?? , in 
general, a ?? -field may not contain all subsets of ?? . _ 
Example 2.1 
(i) F = ?? ,??  is a sigma field, called the trivial sigma-field; 
(ii) Suppose that ?? ??? . Then F = ?? ,?? ?? ,?? ,??  is a ?? -field of subsets of ?? . It is 
the smallest sigma-field containing the set ?? ; 
(iii) Arbitrary intersection of ?? -fields is a ?? -field (see Problem 3 (i));  
(iv) Let ?? be a class of subsets of ?? and let  ?? ?? :?? ???  be the collection of all ?? -
fields that contain ?? . Then 
                                                F = F
?? ?? ??? 
is a ?? -field and it is the smallest ?? -field that contains class ?? (called the ?? -
field generated by ?? and is denoted by ?? (?? )) (see Problem 3 (iii)); 
(v) Let ?? =R and let ?? be the class of all open intervals in R. Then B
1
=?? ??  is 
called the Borel ?? -field on R. The Borel ?? -field in R
?? (denoted by B
?? ) is the 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   3 
 
?? -field  generated by class of all open rectangles in R
?? . A set ?? ?B
?? is called 
a Borel set in R
?? ; here R
?? = {(?? 1
,… ,?? ?? ):-8 <?? ?? < 8, ?? = 1,… ,?? }  
denotes the ?? -dimensional Euclidean space; 
(vi) B
1
 contains all singletons and hence all countable subsets of  
 R  ?? =  ?? -
1
?? ,?? +
1
?? 
8
?? =1
  . _ 
Let ?? be an appropriately chosen class of basic subsets of ?? for which the probabilities 
can be assigned to begin with (e.g., if ?? =R then ?? may be class of all open intervals in 
R; if ?? is a countable set then ?? may be class of all singletons  ?? ,?? ??? ). It turns out (a 
topic for an advanced course in probability theory) that, for an appropriately chosen class 
?? of basic sets, the assignment of probabilities that is consistent with properties (i)-(iii) of 
classical (or relative frequency) method can be extended in an unique manner from ?? to 
?? (?? ), the smallest ?? -field containing the class ?? . Therefore, generally the domain F of a 
probability measure is taken to be ?? (?? ), the ?? -field generated by the class ?? of basic 
subsets of ?? .  We have stated before that we will not care about how assignment of 
probabilities to various members of event space F (a ?? -field of subsets of ?? ) is done. 
Rather we will be interested in properties of probability measure defined on event space 
F. 
Let ?? be a sample space associated with a random experiment and let F be the event 
space (a ?? -field of subsets of ?? ). Recall that members of F are called events. Now we 
provide a mathematical definition of probability based on a set of axioms. 
Definition 2.3 
(i) Let F be a ?? -field of subsets of ?? . A probability function (or a probability 
measure) is a set function ?? , defined on F, satisfying the following three axioms: 
 
(a) ?? ?? = 0,   ??? ?F;                         (Axiom 1: Non- negativity);                         
(b) If ?? 1
,?? 2
,… is a countably infinite collection of mutually exclusive events 
 i. e., ?? ?? ?F, ?? = 1, 2,… ,?? ?? n?? ?? =?? ,?? ???  then 
??  ?? ?? 8
?? =1
 = ?? ?? ?? 
8
1=1
;               (Axiom 2: Countably infinite additive)  
(c) ?? ?? = 1             (Axiom 3: Probability of the sample space is 1). 
 
(ii) The triplet  ?? ,F,??  is called a probability space. _ 
 
 
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   4 
 
Remark 2.2 
(i) Note that if ?? 1
,?? 2
,… is a countably infinite collection of sets in a ?? -field F then 
 ?? ?? 8
?? =1
?F and, therefore, ?? ( ?? ?? 8
?? =1
) is well defined; 
(ii) In any probability space  ?? ,F,??  we have ?? ?? = 1 (or ?? ?? = 0; see Theorem 
2.1 (i) proved later) but if ?? ?? = 1 (or ?? ?? = 0), for some ?? ?F, then it does 
not mean that ?? =?? ( or ?? =?? ) (see Problem 14 (ii)). 
(iii) In general not all subsets of ?? are events, i.e., not all subsets of ?? are elements of 
F. 
(iv)  When ?? is countable it is possible to assign probabilities to all subsets of ?? using 
Axiom 2 provided we can assign probabilities to singleton subsets  ??  of ?? . To 
illustrate this let ?? = ?? 1
,?? 2
,…   or ?? = ?? 1
,… ,?? ?? , for some n ?N  and let 
??  ?? ??  =?? ?? , ?? = 1, 2,…, so that 0=?? ?? = 1, ?? = 1,2,… (see Theorem 2.1 (iii) 
below) and  ?? ?? =
8
?? =1
 ??  ?? ??  
8
?? =1
=??   ?? ?? 
8
?? =1
 =?? ?? = 1. Then, for any 
?? ??? , 
?? ?? =  ?? ?? .
?? :?? ?? ??? 
Thus in this case we may take F =?? ?? , the power set of ?? . It is worth 
mentioning here that if ?? is countable and ?? =  ?? : ?? ???  (class of all 
singleton subsets of ?? ) is the class of basic sets for which the assignment of the 
probabilities can be done, to begin with, then ?? (?? ) =?? ??  (see Problem 5 (ii)).  
(v) Due to some inconsistency problems, assignment of probabilities for all subsets of 
?? is not possible when ?? is continuum (e.g., if ?? contains an interval). _ 
Theorem 2.1 
Let  ?? ,F,??  be a probability space. Then 
(i) ?? ?? = 0; 
(ii) ?? ?? ?F,?? = 1, 2,… .?? , and ?? ?? n?? ?? =?? , ?? ??? ???  ?? ?? ?? ?? =1
 = ?? ?? ?? 
?? ?? =1
 
(finite additivity); 
(iii) ??? ?F, 0=?? ?? = 1 and ?? ?? ?? = 1-?? ?? ; 
(iv) ?? 1
,?? 2
?F and?? 1
??? 2
??? ?? 2
-?? 1
 =?? ?? 2
 -?? ?? 1
 and?? ?? 1
 =?? ?? 2
  
(monotonicity of probability measures); 
(v) ?? 1
,?? 2
?F??? ?? 1
??? 2
 =?? ?? 1
 +?? ?? 2
 -?? ?? 1
n?? 2
 . 
Proof. 
(i) Let ?? 1
=?? and ?? ?? =?? , ?? = 2, 3,…. Then ?? ?? 1
 = 1, (Axiom 3), ?? ?? ?F, ?? =
1, 2,… , ?? 1
= ?? ?? 8
?? =1
 and ?? ?? n?? ?? =?? , ?? ??? . Therefore,  
 
NPTEL- Probability and Distributions 
 
Dept. of Mathematics and Statistics Indian Institute of Technology, Kanpur                                   5 
 
                                                       1 =?? ?? 1
 =??  ?? ?? 8
?? =1
  
 
                                                          = ?? ?? ??                 (using Axiom 2)
8
?? =1
 
 
              = 1 + ?? ?? 
8
?? =2
 
 
               ? ?? ?? 
8
?? =2
= 0 
 
        ??? ?? = 0. 
 
(ii) Let ?? ?? =?? , ?? =?? + 1, ?? + 2,… . Then?? ?? ?F, ?? = 1, 2,… ,?? ?? n?? ?? =?? ,?? ??? and 
?? ?? ?? = 0, ?? =?? + 1,?? + 2,…. Therefore,  
??  ?? ?? ?? ?? =1
 =??  ?? ?? 8
?? =1
              
                                              = ?? ?? ??          using Axiom 2 
8
?? =1
 
                              = ?? ?? ?? 
?? ?? =1
.                     
(iii) Let ?? ?F. Then ?? =?? ??? ?? and ?? n?? ?? =?? . Therefore 
                 1 =?? ??  
                    =?? ?? ??? ??  
=?? ?? +?? ?? ??  (using (ii))         
??? ?? = 1 and ?? ?? ??  = 1-?? ??     (since ?? (?? ?? )? [0,1]) 
? 0=?? ?? = 1 and ?? ?? ?? = 1-?? ?? . 
(iv) Let ?? 1
,?? 2
?F and let?? 1
??? 2
. Then ?? 2
-?? 1
?F,?? 2
=?? 1
? ?? 2
-?? 1
 and 
?? 1
n ?? 2
-?? 1
 =?? . 
 
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