If $g$ nonnegative has bounded support and $\int g^2 d \lambda$ is finite then $\int g d \lambda$ is finite

Question

If $g$ nonnegative has bounded support and $\int g^2 d \lambda$ is finite then $\int g d \lambda$ is finite

Previous question asked you to prove Markovs inequality so I think it may have something to do with that. I was thinking of showing that $\infty > \int g^2 d \lambda > \int g d \lambda$ but this I realized isn't necessarily true and now I'm all out of tactics.

Anyone know how to go about this question?

Thanks

Answer 1

Use that $g\leqslant1+g^2$ on the bounded interval $I$ mentioned in the question and $g=0$ elsewhere, hence $$\int_\mathbb Rg\mathrm d\lambda\leqslant\lambda(I)+\int_\mathbb Rg^2\mathrm d\lambda.$$