Some questions on the proof of Hoelders inequality.

Question

I have some questions about the proof of Hoelder's inequality.

Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in L_q(X)$. Then $fg \in L_1(X)$ and $\|fg\|_1 \le \|f\|_p \|g\|_q$.

Proof: Let $f \in L_p$ and $g \in L_q$. If either $\|f\|_p = 0$ or $\|g\|_q = 0$ then $fg = 0$ $\mu$-almost everywhere and the inequality follows. So we assume $\|f\|_p \ne 0$ and $\|g\|_q \ne 0$. We now let $x \in X$, $A = \frac{|f(x)|}{\|f\|_p}$ and $B = \frac{|g(x)|}{\|g\|_q}$. We can now apply Young's inequality for $A$ and $B$ to get that \begin{align*} \frac{|f(x)||g(x)|}{\|f\|_p \|g\|_q} \le \frac{|f(x)|^p}{p \|f\|_p^p} + \frac{|g(x)|^q}{q\|g\|_q^q}. \end{align*} Since $f^p$ and $g^q$ are integrable it follows that $fg$ is integrable and so $fg \in L_1(X)$. If we integrate both sides with respect to $\mu$ we get \begin{align*} \frac{1}{\|f\|_p \|g\|_q} \int |f(x)g(x)| \, d\mu(x) \le \frac{1}{p} + \frac{1}{q} = 1 \Longrightarrow \|fg\|_1 \le \|f\|_p \|g\|_q. \end{align*}

The following implications I don't understand:

• If either $\|f\|_p = 0$ or $\|g\|_q = 0$ then $fg = 0$ $\mu$-almost everywhere.
• $f^p$ and $g^q$ are integrable. Do we know this because we know that $f$ and $g$ are integrable?
• Since $f^p$ and $g^q$ are integrable it follows that $fg$ is integrable.

Thanks in advance.

Answer 1

• That $\int f\,d\mu=0$ with $f\ge0$ implies $f=0$ a.e. is very basic fare in integration theory. Your first point follows by applying this to either $|f^p|$ or $|g^q|$.
• That $f^p$ and $g^q$ are integrable is the definition of $L^p$ and $L^q$.
• The integrability of those two shows that the integral of the right of your inequality is finite. Hence the integral of the left hand side is finite, and you're done.