Hölder's Inequality for Integrals/General

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

For $i = 1, \ldots, n$ let $p_i \in \R_{>0}$ such that:
 * $\ds \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$

Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes Lebesgue space.

Then their pointwise product $\ds \prod_{i \mathop = 1}^n f_i$ is integrable, that is:
 * $\ds \prod_{i \mathop = 1}^n f_i \in \map {\LL^1} \mu$

and:


 * $\ds \norm {\prod_{i \mathop = 1}^n f_i}_1 = \int \size {\prod_{i \mathop = 1}^n f_i} \rd \mu \le \prod_{i \mathop = 1}^n \norm {f_i}_{p_i}$

where the various instances of $\norm {\, \cdot \,}$ signify $p$-seminorms.

Proof
We use the Principle of Mathematical Induction.

Let it be assumed that the result holds for $i = n - 1$.

We show that the result holds for $i = n$.

Define:
 * $q_n := \dfrac {p_n} {p_n - 1}$

and for $i = 1, \ldots, n - 1$, define:
 * $r_i := p_i \cdot \paren {1 - \dfrac 1 {p_n} }$

Then:
 * $\dfrac 1 {p_n} + \dfrac 1 {q_n} = 1$
 * $\ds \sum_{i \mathop = 1}^{n - 1} \dfrac 1 {r_i} = 1$

and:
 * $q_n \cdot r_i = p_i$

Applying Hölder's Inequality for Integrals to $\ds f := \prod_{i \mathop = 1}^{n - 1} f_i$ and $g := f_n$, we find: