Pulling out a maximum term from an integral

Question

Suppose $g$ is nonnegative on $[0,1]$. Is the following true?

$$ \left| \int_0^1 f(x) g(x) dx\right| \leq \|g\|_\infty \left| \int_0^1 f(x) dx\right| $$

This would be like Holder's inequality, except without pulling the absolute value into the integral of $f$.

Answer 1

Let $g = \mathbf{1}_{[0, 1/2]}$ and $f = \mathbf{1}_{[0, 1/2]} - \mathbf{1}_{(1/2, 1]}$. Then the left-hand side is $1/2$ while the right-hand side is $1 \cdot 0 = 0$.