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Convergence a.e. and of norms implies that in Lebesgue space
I am trying to show that if $ \int_X |f_n|d\mu \to \int_X|f|d\mu $ where $f$ and all the $f_n$ have finite integral and $f_n \to f$ pointwise, then $ \int_X |f_n-f|d\mu \to 0. $
I worked out a proof in the case that $\mu(X) < \infty$, but it relies on Egoroff's theorem which may fail if $\mu(X) = \infty$. I can't find a counterexample in the case $\mu(X) = \infty$ but I suspect that it may not be true. I was thinking of $X=\mathbb{R}$ but maybe there is a good counting measure counterexample on $\mathbb{N}$.
Does anyone know if this is true in the case $\mu(X) = \infty$, and if so, how might I get started in showing it?