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I am studying measure theory myself. here are two problems I tried but failed to solve.

The first one:

Let $A$ be a measurable set on $[0,1]$. Prove that the set $B = \{x^2 \mid x \in A \}$ is measurable as well and $m(B) \le 2m(A)$.

I was thinking to use a continuous mapping of the two sets, but did not know any useful theorem.

The second one:

Let $A$ and $B$ be two closed bounded sets on the line. Prove that the set $A+B=\{x+y \mid x \in A, y \in B\}$ is closed and bounded as well, and $m(A+B) \ge mA +mB$.

Can anyone give a complete solution to any of these two? I'd appreciate your help.

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    Hint for first one: try doing the case where $A$ is an interval. Hint for second one: "compact"2011-10-24
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    Look at Nick's answer here http://math.stackexchange.com/questions/73546/sum-of-two-closed-sets-is-measurable2011-10-24
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    @GEdgar, could you tell me more about the first one?2011-10-24

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