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Let $A$ be a measurable subset of $\Bbb R$ and $\lambda(A)=1$ where $\lambda $ denotes the Lebesgue Measure on $\Bbb R$ .

Prove that there exists a Lebesgue Measurable subset $B$ of $A$ such that $\lambda(B)=\frac{1}{2}$.

I have completed sections on Measure Theory from G.D.Baraa upto Lebesgue Measure but I am not getting any hints on how to solve this problem.Please give me some hints.

Also please give me some names of books where I can find these sort of questions and try to solve them.

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The function $$ F(x) = \int_{-\infty}^{x}\mathbb{1}_A(t)\,dt $$ is absolutely continuous and ranges from $0$ to $1$ since $\mu(A)=1$. By the intermediate value property of continuous functions we have that for some $x_0\in\mathbb{R}$ $$ F(x_0)=\frac{1}{2} $$ holds, hence $\mu\left(A\cap(-\infty,x_0]\right)=\frac{1}{2}$ as wanted.

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  • $\begingroup$ Two questions: Why does $F$ range from $0$ to $1$? $\endgroup$ – Learnmore Dec 19 '17 at 14:08
  • $\begingroup$ Why is $\mu (A\cap (-\infty,x_0])=\frac{1}{2}$ $\endgroup$ – Learnmore Dec 19 '17 at 14:09
  • $\begingroup$ Can you suggest any book where these type of questions are discussed $\endgroup$ – Learnmore Dec 19 '17 at 14:10

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