Help with a real analysis problem

Thread starter polkadot66
Start date Aug 9, 2019
Tags

Analysis Real analysis

Click For Summary

SUMMARY

The discussion centers on proving the existence of a measure ## \mu'## in the context of real analysis, specifically involving functions in L^1. The participant emphasizes starting with the definition of L^1 functions to construct a sequence of step functions that approximate the function f. This approach allows for the definition of sets of finite measure to modify the measure mu directly, rather than using a proof by contradiction.

PREREQUISITES

Understanding of L^1 functions in real analysis
Familiarity with measure theory concepts
Knowledge of step functions and their properties
Experience with proof techniques in mathematics

NEXT STEPS

Study the properties of L^1 functions and their applications in measure theory
Learn about constructing sequences of step functions for function approximation
Research the modification of measures and finite measure sets
Explore direct proof techniques in mathematical analysis

USEFUL FOR

Mathematics students, educators, and researchers interested in real analysis and measure theory, particularly those focusing on function approximation and measure modification techniques.

Aug 9, 2019

polkadot66

Homework Statement: Given ## f \in L^1(X, M, \mu)## , show there is a ## \sigma## -finite measure ## \mu'## such that ## \int_{E} f d\mu = \int_{E} f d\mu'## .

Relevant Equations: ## f \in L^1## so ## \int |f| d\mu < \infty##
a measure ## \mu'## is ## \sigma##-finite if there are sets ##A_1, A_2, ...## such that ## \mu'(A_n) < \infty## and ##X= \cup A_n##

I tried to prove this by absurd stating that there is no such ## \mu'## but i couldn't get anywhere...

Physics news on Phys.org

Aug 22, 2019

mathwonk

Science Advisor

Homework Helper

i would begin with the definition of f being in L^1. That should give you a sequence of step functions that approximate f and can be used to define some sets of finite measure that can be used to modify mu. anyway i would do it directly, not by contradiction.