Problem in showing convergence in almost uniformly on finite measure set implies Convergence in $L^2$ measure

Question

$(X,M,\mu)$ is measure space with $\mu(X)<\infty$ if $f_n\to f$ uniformly almost everywhere then I have to show that $f_n\to f$ converges in $L^2$

$X=A\cup A^c$ where on A $f_n\to f$ uniformly and $\mu(A^c)=0$

I can control part of A but on $A^c$ I can not do any thing as $\sup|f_n-f|^2$ may not exist . how to tackle this problem ?

ANy help will be appreciated

Answer 1

You can't show $f_n\to f$ in $L^2,$ because $f_n,f$ may not belong to $L^2.$ Example: $f_n(x)=1/x+1/n$ on $X=(0,1)$ with Lebesgue measure. However, showing $\int_X|f_n-f|^2\,d\mu \to 0$ is simple with your hypotheses (assuming the measurability of these functions), as the comment by @lc2r43 indicates.