Problem about convergence of an integral

Question

Assume that we have a sequence $f_k \in L^1{(\Bbb R^n)}$ pointwise converging to $f$ such that

1. $\left|f_k\right|$ converges pointwise and monotonically to $\left| f \right|$
2. The limit $\lim_k\int_{\Bbb R^n} f_k$ exists finite

Can we conclude that $f$ is in $L^1{(\Bbb R^n)}$? Or, at least, can we say something about the equality $\lim_k \int f_k = \int f$? I have no idea how to do this, thanks in advance to everyone that is willing to help me!

Answer 1

Let $f_k(x)=x$ for $|x| \leq k$ and $0$ for $|x| >k$. Let $f(x)=x$ for all $x$. Then your hypothesis is satisfied (with $\int f_k=0$ for all $k$) but $f$ is not integrable.