Proving inclusion $L^q \subseteq L^p$ for $1 \leq p \leq q \leq \infty$

Question

I want to make sure my proof is correct.

Problem:

Let $(X,\mathcal{A},\mu)$ be some measure space.

If $1 \leq p \leq q \leq \infty$, and $\mu(X) < \infty$, then $L^q \subseteq L^p$

Case 1: Let $q = \infty$.

If $f \in L^\infty$, then there is a number $M: |f(x)| \leq M \text{ a.e.}$ on $x$.

Therefore, $|f|^p \leq M^p \text{ a.e.}$

and

$$||f||_p^p = \int_X |f|^p d\mu \leq \int_X M^p d\mu = M^p \mu(X) < \infty$$

Therefore, $f \in L^p$ and $L^\infty \subseteq L^p$

Now, let $1 \leq q < \infty$

Partition the set $X$ into

$$A := \{x: |f(x)|<1\}$$ $$B :=\{x:|f(x)| \geq1\}$$

Then,

$$\int_X |f|^p d\mu = \int_A |f|^p d\mu + \int_B |f|^p d\mu \leq \int_A |f|^p d\mu + \int_Bd\mu$$

But,

$$\int_Bd\mu = \mu(B) \leq \mu(X) < \infty$$

and

$$\int_A |f|^p d\mu \leq \int_A |f|^q d\mu\leq \int_X|f|^q d\mu = ||f||_q^q<\infty$$

since $f \in L^q$ by hypothesis.

Therefore, $\int_X|f|^p d\mu < \infty$ and $f \in L^p$

Which entails

$$L^q \subseteq L^p$$

Answer 1

• Your definition of $A$ and $B$ should be swapped.
• Your separation of case 1 and case 2 is fine. But I think case 1 is unnecessary and can be deleted as it can be part of case 2.

Other than the above notes, your answer looked fine to me.

Answer 2

Expanding the previous answer, notice that your proof gives no estimate about the relationship between the two norms. Using instead Hölder's inequality you can show that the inclusions are Lipschitz continuous, with Lipschitz constant depending on $\mu(X)$.