Does $\int 1_{|m|>A} m^2$ converge to zero for an $L^2$ function?

Question

We assume that $m \in L^2(0,1)$, hence $\int_0^1 m(x)^2 dx< \infty$ but that $m \not \in L^\infty(0,1)$.

Hence $\{ x: |m(x)|>A \}$ has always positive measure.

Now the question:

Does $\int 1_{|m(x)|>A} m(x)^2 dx \rightarrow 0$ for $A \rightarrow \infty$?

Answer 1

$$1_{|m(x)|>1}m^2(x)\leq m^2(x)\in L^1$$ therefore by the dominated convergence theorem $$\lim_{A\to\infty }\int 1_{|m(x)|>A}m(x)^2dx=\int \underbrace{\lim_{A\to\infty }1_{|m(x)|>1}m(x)^2dx}_{=0}=0$$