Property of a limit of functions of average value zero in L^2 space

[lmedin02]

Homework Statement

Let $f_k\rightarrow f$ in $L^2(\Omega)$ where $|\Omega|$ is finite. If $\int_{\Omega}{f_k(x)}dx=0$ for all $k=1,2,3,\ldots$, then $\int_{\Omega}{f(x)}dx=0$.

Homework Equations

The Attempt at a Solution

I started by playing around with Holder's inequality and constructing examples where this is the case. Since every other example I create usually does not converge to a function. I used functions defined on a bounded symmetric interval like $\dfrac{1}{k}x$, $\sin(\dfrac{1}{k}x)$. However, I do not see how to actually set up to make this conclusion.

 

[jbunniii]

What if you write
$$\begin{align} \left| \int_{\Omega} f(x) dx \right| &=
\left| \int_{\Omega} f(x) dx - \int_{\Omega} f_k(x) dx \right| \\
&= \left|\int_{\Omega} (f(x) - f_k(x)) dx \right| \\
\end{align}$$
and apply an appropriate inequality to the right hand side?

 

[lmedin02]

Got it. Thanks. Using Holder's inequality and noting that the size of $\Omega$ is finite the results follows since $f_k\rightarrow f$ in $L^2(\Omega)$.