Hölder's Inequality for Integrals/Equality

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p, q \in \R_{>0}$ such that $\dfrac 1 p + \dfrac 1 q = 1$.

Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space.

Then equality in Hölder's Inequality for Integrals, that is:
 * $\ds \int \size {f g} \rd \mu = \norm f_p \cdot \norm g_q$

holds, for $\mu$-almost all $x \in X$:
 * $\dfrac {\size {\map f x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$


 * $\dfrac {\size {\map f x}^{p - 1} } {\size {\map g x} } = c$

for some $c \in \R$.