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I'm preparing to the second mini-test in measure theory. Here is one of the problems I cannot deal with. I would appreciate any help, thank you.

Let $\mu$ be a Radon measure on $\mathbb{R}$, suppose that $A$ is a $\mu$–measurable subset of $[a,b]$ and let $h$ be a positive number. Prove that $\frac{1}{2h}\int_a^b\mu(A\cap(x-h,x+h))\,\text{d}x\le \mu(A).$

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If you apply Fubini (or Tonelli) in the second equality below (everything is positive) you get $ \frac1{2h}\,\int_{[a,b]}\mu(A\cap(x-h,x+h))\,dx=\frac1{2h}\,\int_{[a,b]}\int_{(x-h,x+h)}\,1_A(t)\,d\mu(t)\,dx\\ \leq\frac1{2h}\int_{(a-h,b+h)}\int_{[t-h,t+h]}\,1_A(t)\,dx\,d\mu(t)=\int_{(a-h,b+h)}\,1_A(t)\,d\mu(t)=\mu(A\cap(a-h,b+h))=\mu(A). $

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    Exactly. You are welcome! – 2012-11-19