Pointwise Convergence, L^2 Convergence I was wondering if pointwise convergence of a sequence of functions implies convergence with respect to the $L^2$ norm. So if $g_n$ is a sequence of functions in $L^2[0,1]$ converging to g pointwise, does it converge to g with respect to the $L^2$ norm? Thank you.
 A: No (in terms of probability spaces for instance, you have the distinction between convergence a.e. and convergence in $L_2$).
Or, look for instance at the function $f_n\colon x\mapsto\mathbb{1}_{[n,n+1]}(x)$. It converges pointwise to $\bar{0}$; yet, the integral $\int_\mathbb{R} f_n^2$ is always equal to 1.
A: The answer is No. 
Like Clement pointed out. However we can add some more condition(s) to make the following claim valid.

Pointwise convergence for $\{f_n\}\subset L^p(X)$ + Condition (A) $\Longrightarrow$ $L^p$ convergence $\|f_n -f \|_{L^p} \to 0$.

where $1\leq p <\infty$, when $p = \infty$, the situation gets more complicated.
(A) can be


*

*$\|f_n\|_{L^p}\to \|f\|_{L^p}$. This rules out the possibility that $f_n = \chi_{[n,n+1)}(x)$ in Clement's counterexample, in which $X$ is not bounded. Also adding this condition rules out another type of counterexample, in which $X=[0,1]$ is bounded and closed: 
$$f_1 = 2^{1/p}\chi_{[0,1/2)}, f_2 = 2^{1/p}\chi_{[1/2,1)}, f_3 = 2^{2/p}\chi_{[0,1/2^2)}, f_4 = 2^{2/p}\chi_{[1/2^2,2/2^2)},\ldots $$ This extra condition normally accompanies the condition $\Big\{\|f_n\|_{L^p}\Big\}$ is bounded, which itself is not enough. But the once the boundedness of the sequence is added, we do have weak convergence meaning $\displaystyle \int f_n g \to \int f g$ for any $g\in L^q$.

*$\Big\{\|f_n\|_{L^p}\Big\}$ is bounded, and $f_n \to f$ in $L^1$. 

*$X$ is compact and $\{f_n\}$ is equicontinuous. The compactness of $X$ rules out the following counterexample: $f_n = n^{-1/p}\chi_{[-n,n]}(x)$. 
Alternatively, we could modify the statement to be:

For $\forall f\in L^p(X)$, there exists $\{f_n\}\subset L^p(X)$ such that $f_n\to f$ in $L^p$ also pointwisely.

To sum up, the topology induced by the norm metric does not have the topology of pointwise convergence (or a.e. convergence). 
