$L_p(\mu)\subseteq L_q(\mu)$

Question

Given a measure space $(\Omega,\mathfrak A,\mu)$ and $1≤q≤p$, how can I show that $$L_p(\mu)\subseteq L_q(\mu)$$ if the measure $\mu$ is finite, that means $\mu(\Omega)<\infty$?

Hint/Solution: Hölder inequality – b00n heT May 25 '16 at 14:18 — b00n heT, May 25 '16 at 14:18
Possible duplicate of $L^p$ and $L^q$ space inclusion – Alex M. May 25 '16 at 14:25 — Alex M., May 25 '16 at 14:25

score 2 · Answer 1 · answered May 25 '16 at 14:23

2

Hint:

$$|f(x)|^q \le \max(|f(x)|^p,1)$$

answered May 25 '16 at 14:23

ajotatxe

65,084

$L_p(\mu)\subseteq L_q(\mu)$

1 Answers1