# Measure Theory Problem

• Oct 5th 2010, 07:43 PM
everk
Measure Theory Problem
Hi everyone
I cannot find the solution to this problem,
Let A a Lebesgue Measurable Set; Prove that given $\displaystyle 0 \leq b \leq m(A)$ then exists $\displaystyle B \subseteq A$ Lebesgue Measurable such that m(B) = b

Thanks
everk.
• Oct 6th 2010, 10:13 AM
Opalg
Quote:

Originally Posted by everk
Let A a Lebesgue Measurable Set; Prove that given $\displaystyle 0 \leq b \leq m(A)$ then exists $\displaystyle B \subseteq A$ Lebesgue Measurable such that m(B) = b

Let $\displaystyle {\mathstrut\chi}_A$ denote the characteristic function of $\displaystyle A$ (so that $\displaystyle {\chi\mathstrut}_A(t) =1$ if $\displaystyle t\in A$, and $\displaystyle {\mathstrut\chi}_A(t) = 0$ if $\displaystyle t\notin A$). For each real number $\displaystyle x$, let $\displaystyle \displaystyle f(x) = \int_{-\infty}^x\!\!{\mathstrut\chi}_A(t)\,dt = m(A\cap(-\infty,x]).$ Show that $\displaystyle f$ is continuous and use the intermediate value theorem to deduce that there exists $\displaystyle c$ such that $\displaystyle f(c)=b.$
• Oct 6th 2010, 11:36 AM
Bruno J.
Does that work if $\displaystyle A=(-\infty, 0)$?
• Oct 6th 2010, 12:10 PM
Opalg
Quote:

Originally Posted by Bruno J.
Does that work if $\displaystyle A=(-\infty, 0)$?

It will certainly need a bit of adjustment if $\displaystyle m(A) = \infty$. I think that you would then have to start by looking at $\displaystyle \displaystyle\int_x^X\!\!{\mathstrut\chi}_A(t)\,dt$, which can be made arbitrarily large if $\displaystyle x$ is small enough and $\displaystyle X$ is large enough. Then reduce $\displaystyle X$ until the integral reaches the value $\displaystyle b$.