If $f_n \to f$ in $L^1$ and $\chi_Af_n,\chi_Af \in L^1$ does $\chi_Af_n \to \chi_Af$ in $L^1$?

Question

This question came up when trying to prove a weaker statement that: $\lim_{n \to \infty}\int f_n=\int f$, then $\lim_{n \to \infty}\int \chi_Af_n=\int \chi_Af$.

Which I think I can proven using Fatous Lemma. Is the statement in the title true? What would be the proof?

Answer 1

Let's assume that $f_{n} \to f$ in $L^{1}(X)$ and $A \subset X$, then \begin{eqnarray} \int_{A}{|f_{n}(x) - f(x)|}dx &=& \int_{X}{|\chi_{A}(x)||f_{n}(x) - f(x)|} dx \\ & \leq & \int_{X}{|f_{n}(x) - f(x)|}dx \to 0. \end{eqnarray}