Hölder's Inequality for Integrals/Equality

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 equality in Hölder's Inequality, that is:
 * $\displaystyle \int \left\vert{f g}\right\vert \, \mathrm d \mu = \left\Vert{f}\right\Vert_p \cdot \left\Vert{g}\right\Vert_q$

holds, for almost all $x \in X$:
 * $\dfrac {\left\vert{f \left({x}\right)}\right\vert^p} {\left\Vert{f}\right\Vert_p^p} = \dfrac {\left\vert{g \left({x}\right)}\right\vert^q} {\left\Vert{g}\right\Vert_q^q}$