Page 173 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 14: Radon-Nikodym Theorem

CONTENTS
Objectives
Introduction

14.1 Radon-Nikodym Theorem
14.1.1 Absolutely Continuous Measure Function
14.2 Summary

14.3 Keywords
14.4 Review Questions
14.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define Absolutely continuous measure function
 State Radon-Nikodym theorem
 Understand the proof of Radon-Nikodym theorem

 Solve problems on this theorem
Introduction

In mathematics, the Radon-Nikodym theorem is a result in measure theory that states that given
a measurable space (X, ), if a -finite measure on (X, ) is absolutely continuous with respect to
a -finite measure on (X, ), then there is a measurable function f on X and taking values in [0, ],
such that for any measurable set A.
The theorem is named after Johann Radon, who proved the theorem for the special case where
N
the underlying space is R in 1913, and for Otto Nikodym who proved the general case in 1930.
In 1936 Hans Freudenthal further generalised the Radon-Nikodym theorem by proving the
Freudenthal spectral theorem, a result in Riesz space theory, which contains the Radon-Nikodym
theorem as a special case.

If Y is a Banach space and the generalisation of the Radon-Nikodym theorem also holds for
functions with values in Y, then Y is said to have the Radon-Nikodym property. All Hibert
spaces have the Radon-Nikodym property.
14.1 Radon-Nikodym Theorem

14.1.1 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  .

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