**Definition** (*Pointwise convergence of sets*): We say that a sequence $E\_n$ of sets in $\mathbb{R}^d$ converges pointwise to another set $E$ in $\mathbb{R}^d$ if the indicator functions $1\_{E\_n}$ converge pointwise to the indicator function $1\_E$.

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> The problem is to show that if $E\_n$ are all Lebesgue measurable, and converge pointwise to $E$, then $E$ is also Lebesgue measurable.
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> Further, I need to show that $m(E\_n)$ converges to $m(E)$ [where $m(\cdot)$ denotes the Lebesgue measure], if $E\_n$ are all contained in another Lebesgue measurable set $F$ of finite measure. Also, I've to produce a counter example to show that the convergence may not hold without the assumption.
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For the first part, I tried to write $E$ as some countable union/intersection of the $E\_n$'s. But I cannot exploit the definition of pointwise convergence of sets well enough to defend my point. For the second part, probably I need to incorporate the monotone convergence theorem, but I don't have a clear-cut attack on the problem. Any help would be greatly appreciated!