Suppose $\mu(X)=1$ and $f, g $are nonnegative function such that $fg \geq 1$ a.e. prove $(\int fd\mu)(\int gd\mu) \geq 1$.

Question

I have a problem in $L^p$ space. Suppose $\mu(X)=1$ and $f, g $are nonnegative function such that $fg \geq 1$ a.e. prove $(\int fd\mu)(\int gd\mu) \geq 1$.

I have no any idea to prove that. Holder inequality? thanks

Answer 1

Using Cauchy-Schwarz, you have $$\left(\int \sqrt{f}^2 \right)^{\frac{1}{2}} \left( \int \sqrt{g}^2 \right)^{\frac{1}{2}} \geq \int \sqrt{fg} \geq \int 1 = 1$$

So, squaring both sides, you get the result.

Answer 2

Taking the square root of the inequality gives $$\sqrt f \sqrt g \geq 1.$$ can you see how to proceed?