"Proving" $L^q \subseteq L^p$ for any $p <\infty$.

Question

In Folland's Real Analysis, he gives the following proof (Proposition 6.12) that $L^q \subseteq L^p$ when $0 \leq p < q \leq \infty$. My question is, nowhere in the proof is the assumption that $p < q$ used. Couldn't this proof be used to "prove" that the claim holds even if $q < p$ (which I know is not true, by counterexamples)? Where would that proof breakdown?

Answer 1

The proof relies on applying Hölder's inequality with the two conjugate exponents: $r=q/p$ and $r' = q/(q-p)$. Indeed, $1/r + 1/r' = 1$, so it seems like we can use Hölder's inequality. However, Hölder also requires that both $r$ and $r'$ are positive, which is not satisfied for this choice of exponents if $p > q$.

Answer 2

Here is an elementary argument not based on the Hölder inequality. Let $f\in L^q$ and $$A=\{x\in X\,:\,|f(x)|\ge 1\}$$ Then for $q>p$ we get $$\int\limits_X|f(x)|^p\,d\mu(x)=\int\limits_A|f(x)|^p\,d\mu(x)+\int\limits_{X\setminus A}|f(x)|^p\,d\mu(x)\\ \le \int\limits_A|f(x)|^q\,d\mu(x)+\mu(X\setminus A)<\infty $$