Exercise 6.L - The Elements of Integration and Lebesgue Measure by Bartle

Question

$\textbf{6.L}$ Suppose that $X = \mathbb{N}$ and $\mu$ is the counting measure on $\mathbb{N}$. If $f \in L_p$, then $f \in L_s$ with $1 \leq p \leq s < \infty$ and $||f||_s \leq ||f||_p$.

This is what I thought:

I know that $\int_X |f|^p d\mu < \infty$, because $f \in L_p$.

Define $A := \{ x \in X \ ; \ |f(x)| \geq 1 \}$, so $\int_{A^c} |f|^s d\mu \leq \int_{A^c} |f|^p d\mu \leq \int_X |f|^p d\mu < \infty$. This imply that $f \chi_{A^c} \in L_s$. Furthermore, $\int_A |f|^p d\mu \leq \int_A |f|^s d\mu$, but I don't know what I need to do now. I think that I need use the fact that $\mu$ is the counting measure on $\mathbb{N}$. I would like to a hint.

Answer 1

Let $f\in L_p$ where $f:\Bbb N\rightarrow \Bbb R$ , define $g=\frac{f}{||f||_p}$ then $||g||_p=1$ (note that when $||f||_p=0$ then $f=0$ and then all facts hold trivially). Write $g=\{a_1,a_2,...\}$ then $$\int_X |g|^p d\mu =\int_X \sum_{n=1}^\infty|g|^p\chi_{\{n\}}\ d\mu=\sum_{n=1}^\infty \int_X |g|^p\chi_{\{n\}}\ d\mu =\sum_{n=1}^\infty |a_n|^p=1$$ Therefore we have $|a_n|≤1$ for each $n≥1$. So that $|a_n|^p≥|a_n|^s$ i.e. $1=||g||_p^p≥||g||_s^p$ which gives $||g||_s≤1$ i.e. $||f||_s≤||f||_p$.

Use monotone convergence to prove interchangeability of integral and summation for sequence of non-negative functions.

Answer 2

The fact that it's the counting measure is key, in fact if we take $X = (0, 1)$ with the Lebesgue measure the opposite result holds.

My hint would be WLOG by rescaling that $\lVert f \rVert_s = 1$. Now what happens to the set $A$ you defined?