According to Wikipedia one can define the arithmetic mean of an integrable function $f: M \to \mathbb{R}$ over a relatively compact domain $M$ as:
$$A(f) = \frac{1}{\mu(M)}\int_M f$$
where $\mu(M)$ is the measure of $M$. Moreover, if $f > 0$ one can also define its geometric mean as:
$$G(f) = \exp\left(\frac{1}{\mu(M)}\int_M \log f\right)$$
The inequality of the arithmetic and geometric means is a well-known result in the discrete case. ¿Is it also true in this context? That is, is it true that:
$$A(f) \geq G(f)$$
for every positive integrable function $f$?