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).