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.