How can I prove $f=g$ when $f_k$ pointwisely converges to $f$, $L^2$ converges to $g$, and $f$, $g$ are continuous?

Question

Assume that ${f_k}$ is a sequence of integrable functions on $[a,b]$ that converges pointwise to $f$ on $[a,b]$ and that converges in the mean to $g$ on $[a,b]$. Assume also that both $f$ and $g$ are continuous. Then how can I get $f=g$?? Please help me.

Answer 1

Here's the special case when $[a,b]=[0,1]$, $g=0$, and $f>1$:

Let $\{a_n\}_{n=1}^\infty$ be any sequence of positive numbers with $\sum a_n<1$ (e.g., $a_n=10^{-n}$). Since $f_k$ $L^2$-converges to $g$, we find $k_n$ such that the set $E_n:=\{\,x\in[0,1]\mid f_{k_n}(x)>1\,\}$ has measure $<a_n$; we may assume wlog that $k_1<k_2<k_3<\ldots$. From $\sum \mu(E_n)<1$, we know that there exists $x_0\in[0,1]$ that is in none of the $E_n$. Hence $f_{k_n}(x_0)\le 1$ for all $n$, contradicting $\lim f_k(x_0)=f(x_0)\ge 1$.

You can easily reduce the general case to the above special case by considering a small interval around a point where $f$ and $g$ differ, subtract $g$ from everything, transform, and scale appropriately.