Definition of functions in $L^p$ space

Question

I know that if we suppose that $1 \leq p \leq \infty$, and if $f$ is in $L^p$, then this means that $\|f\|_p=[\int_X (f^p) dx]^{\frac{1}{p}}$.

But I feel as though I'm missing some important implications when we state that $f$ is in $L^p$. What else is implied in the definition of $L^p$?

Actually $f\in L^p$ means that integral is finite. Otherwise all you've stated is the definition of $|\cdot|_p$ — Gregory Grant, Jun 01 '15 at 19:48

Alonso Delfín · Accepted Answer · 2015-06-01T20:01:44.613

2

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. If $X$ is a set, $\mathcal{S}$ a sigma algebra on $X$ and $\mu : \mathcal{S} \to \mathbb{R}^+$ a measure, then, by definition $$ L^p(X, \mathcal{S}, \mu) = \left\{ f:X\to \mathbb{F} : f \text{ is $\mathcal{S}$ measurable and } \int_X |f|^p d\mu < \infty \right\} $$ In that case, if $f \in L^p(X, \mathcal{S}, \mu)$, the $p$-semi norm is $$ \| f \|_p =\left( \int_X |f|^p d\mu \right)^{1/p} < \infty $$ Note that you forgot to consider the modulus $|f|$ in the semi norm

edited Jun 01 '15 at 20:01

answered Jun 01 '15 at 19:52

Alonso Delfín

6,529

@alias it is the field where the functions live, generally $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. I've made an edit and add more details, hope it is clearer – Alonso Delfín Jun 01 '15 at 19:58
Okay, thanks, I hadn't read your comment thoroughly enough. – alias Jun 01 '15 at 19:58
You are welcome! Glad to help ! – Alonso Delfín Jun 01 '15 at 20:04
@Alonso-do you think you can help me with my other question? – alias Jun 01 '15 at 20:53
Sure I can try ! You can ask it in this site and put the link here in the comment for mi to check it ! – Alonso Delfín Jun 01 '15 at 21:50
http://math.stackexchange.com/questions/1308336/lp-space-corollary/1308356#1308356 – alias Jun 01 '15 at 21:57
@alias Done. I have put an answer. Hope it is clear. – Alonso Delfín Jun 01 '15 at 22:33

Definition of functions in $L^p$ space

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