Hölder's Inequality for Integrals

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $p, q \in \R$ such that $\dfrac 1 p + \dfrac 1 q = 1$.

Let $f \in \mathcal{L}^p \left({\mu}\right), f: X \to \R$, and $g \in \mathcal{L}^q \left({\mu}\right), g: X \to \R$, where $\mathcal L$ denotes Lebesgue space.

Then their pointwise product $f g$ is integrable, i.e. $f g \in \mathcal{L}^1 \left({\mu}\right)$, and:


 * $\left\Vert{f g}\right\Vert_1 = \displaystyle \int \left\vert{f g}\right\vert \, \mathrm d \mu \le \left\Vert{f}\right\Vert_p \cdot \left\Vert{g}\right\Vert_q$

where the $\left\Vert{\cdot}\right\Vert$ signify $p$-seminorms.

Equality
Equality, i.e.: