How do I infer that $\lim_{k \rightarrow \infty} \chi_{B_k} =0$?

Question

How do I infer that $\lim_{k \rightarrow \infty} \chi_{B_k} =0$?

I know that $B_k$ measurable, And there's a set inclusion $ ... \subset B_2 \subset B_1$.

And $m(B_k) \rightarrow 0$.

Answer 1

Let $B=\cap_k B_k$. Verify that $\chi_{B_k}(x) \to \chi_B(x)$ for every $x$. Since $0 \leq m(B)\leq m(B_k)$ for every $k$ we get $m(B)=0$ by taking limit as $k \to \infty$. But $m(B)=0$ implies that $\chi_B=0$ almost everywhere. So $\cap_k B_k \to 0$ almost everywhere.