We know that if $f_n$ is a sequence of measurable functions such that $||f_n||_{L^1} < K$, then there exists a subsequence which converges in the weak topology of $M(\Omega)$ (which is the dual of continuous functions with compact support) and further the subsequence converges in the sense of distribution.
Suppose instead of $L^1$ boundedness, if we have $\int_{\mathbb{R}} f_n =K$ for all $n \in\mathbb{N}$ can we say that there exists a subsequence which converges in the sense of distribution. I feel this result should be true, but I am unable to prove it. Please help me to prove or disprove
Say $\int f=0$, and let $f_n=nf$.