Page 179 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
= lim n gd f d as n g fg
g
F F
d d du
= fg .
d d d

14.2 Summary

 Let (X, ) be a measurable space and let r and m be measure functions, defined on the space
(X, A). The measure is said to be absolutely continuous w.r.t. if
(A) = 0 or | | (A) = 0, A  (A) = 0, and is denoted by  .

 Let (X, , ) be a -finite measure space. If Y be a measure defined on A s.t. is absolutely
continuous w.r.t. , then there exists a non-negative measurable function f s.t.

(A) = fd , A 
A
The function f is unique in the sense that if g is any measurable function with the property
defined as above, then f = g almost everywhere with respect to .
14.3 Keywords

Absolutely Continuous Measure Function: Let (X, ) be a measurable space and let and be
measure functions defined on the space (X, A). The measure is said to be absolutely continuous
w.r.t. if
(A) = 0 or | | (A) = 0, A  (A) = 0, and is denoted by  .
Radon-Nikodym Theorem: Let (X, , ) be a -finite measure space. If be a measure defined on
A s.t. is absolutely continuous w.r.t. , then there exists a non-negative measurable function f
on s.t.

(A) = f d , A  .
A
The function f is unique in the sense that if g is any measurable function with the property
defined as above, then f = g almost everywhere with respect to .

14.4 Review Questions

1
d d
1. Show that ,
d d
where and are -finite signed measures and  ,  .

2. If (E) = fd , , where fd exists, then find | | (E).
E E
3. State and prove Radon-Nikodym theorem.

172 LOVELY PROFESSIONAL UNIVERSITY