# Is a bounded integrable function square integrable?

• May 28th 2010, 05:43 PM
aukie
Is a bounded integrable function square integrable?
If $\displaystyle f \in L^1(E)$ is bounded, and E is of finite measure, i.e $\displaystyle m(E)<\infty$, (m here is the lebesgue measure but it can be arbitrary).

Is $\displaystyle f \in L^2(E)$?

Is it enough to say:
since $\displaystyle |f|<M$ for some real M.
$\displaystyle \int _E \left|f\right|^2dm<\int _EM^2dm=M^2\int _Edm=M^2m(E)<\infty$

But what I dont get is that this argument only uses the fact f is bounded, not the fact that f is integrable.

I know that $\displaystyle L^2(E) \subseteq L^1(E)$ since E is of finite measure. I wonder if this has something to do with the answer?
• May 28th 2010, 06:26 PM
aukie
Perhaps if i write $\displaystyle |f|^2=|f||f|<M|f|$ then

$\displaystyle \int _E \left|f\right|^2dm<\int _EM|f|dm=M\int _E|f|dm<\infty$

then it would be true that if f is integrable then it is also square integrable... does this seem plausible?