Show that $(f_{n}g_{n})$ converges to $fg$ in $L^{1}(\\mu)$

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So what we want is that limn ∫|fngn − fg|dμ = 0 lim n ∫ | f n g n f g | d μ = 0 . The condition (ii) (i i) makes me think of Dominated Convergence Theorem, but we don't know if ∫ 2dμ ∫ 2 d μ is integrable so it can't be used directly. Since fn f n converges in mean it does also converge in measure. Can we do something like:

Related Show that $(f_{n}g_{n})$ converges to $fg$ in $L^{1}(\\mu)$