[See Solution] Suppose that E is measurable with m(E)=1. Show that: There is a measurable set F subet; E such that m(F)=1 / 2. [Hint: Consider the function

Question: Suppose that $E$ is measurable with $m(E)=1$. Show that:

There is a measurable set $F \subset E$ such that $m(F)=1 / 2$. [Hint: Consider the function $f(x)=m(E \cap(-\infty, x]))$
There is a closed set $F$, consisting entirely of irrationals, such that $F \subset E$ and $m(F)=1 / 2$.
There is a compact set $F$ with empty interior such that $F \subset E$ and $m(F)=1 / 2$.

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[See Solution] Suppose that E is measurable with m(E)=1. Show that: There is a measurable set F subet; E such that m(F)=1 / 2. [Hint: Consider the function

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