Let $M = (X, \mathcal{F}, \mu)$ be a
D1158: Measure space such that
| (i) |
\begin{equation}
\mu(X)
< \infty
\end{equation}
|
| (ii) |
$f : X \to \mathbb{C}$ is a D5148: Borel-measurable complex function on $M$
|
| (iii) |
$1 \leq q < p \leq \infty$
|
| (iv) |
\begin{equation}
\left( \int_X |f|^p \, d \mu \right)^{1/p}
< \infty
\end{equation}
|
Set
\begin{equation}
r := \frac{p}{q}
\quad \text{ and } \quad
s := \frac{p}{p - q}
\end{equation}
Since $p > q$, then $p - q > 0$ and
\begin{equation}
\frac{1}{r} + \frac{1}{s}
= \frac{q + p - q}{p}
= \frac{p}{p}
= 1
\end{equation}
Consider now the real functions $F, G : X \to \mathbb{R}$ given by
\begin{equation}
F(x) = |f(x)|^q
\quad \text{ and } \quad
G(x) = 1
\end{equation}
Applying
R82: Hölder's inequality with the conjugate exponents $r$ and $s$, one has
\begin{equation}
\begin{split}
\int_X |f|^q \, d \mu
= \Vert F G \Vert_{L^1}
& \leq \Vert F \Vert_{L^r} \Vert G \Vert_{L^s} \\
& = \left( \int_X |f|^{p} \, d \mu \right)^{\frac{q}{p}} \left( \int_X \, d \mu \right)^{\frac{p - q}{p}}
= \Vert F \Vert_{L^p}^q \mu(X)^{\frac{p - q}{p}}
< \infty
\end{split}
\end{equation}
Exponentiating both sides to power $1 / q$, we can then conclude
\begin{equation}
\left( \int_X |f|^q \, d \mu \right)^{1 / q}
< \infty
\end{equation}
$\square$
