This thread is related to: [Spectral Measure: Dominated Convergence](https://math.stackexchange.com/q/1058310/79762)

Given a measure space $\Omega$.

Consider a sequence of square integrables: $\int|f\_n|^2\mathrm{d}\mu<\infty$

Suppose pointwise convergence: $f\_n\to f$.

Does the following hold:
$$\int|f\_m-f\_n|^2\mathrm{d}\mu\to0\implies\int|f-f\_n|^2\mathrm{d}\mu\to0$$

The problem is that they may have no dominant at all:
$$s\_n:=\frac{1}{\sqrt{n}}\chi\_{(n,n+1]}\to0:\quad\int|s\_m-s\_n|^2\mathrm{d}\mu\stackrel{m\neq n}{=}\frac{1}{m}+\frac{1}{n}\to0\quad\int\sup\_n|s\_n|\mathrm{d}\mu=\sum\_{n=1}^\infty\frac{1}{n}=\infty$$