I'm new to measure theory, $L^p$ spaces, etc. I was wondering if my interpretation of why $\|\cdot\|_p$ becomes a norm on $L^p$ is correct.

Consider a measure space $(X,\mathcal{A},\mu)$, and $\mathcal{L}^p = \{f : X \to \mathbb{C}, f \ \text{is measurable}, \|f\|_p < \infty \}$, where $\|f\|_p = \left( \int \lvert f \rvert^p \mathrm{d}\mu \right)^{1/p}$, $p \in [1,\infty)$ is a seminorm on $\mathcal{L}^p$.

Define an equivalence relation on $\mathcal{L}^p$ as $f \sim g \iff f = g, \ \mu \ \text{a.e.}$

Let $L^p$ be the set of equivalence classes. Then $(L^p, \|\cdot\|_p)$ becomes a normed space.

My understanding is that $\|\cdot\|_p$ starts out as a seminorm because it can be $0$ if $f$ is $0 \ \mu$-a.e. For $\|\cdot\|_p$ to be a norm, $\|f\|_p = 0 \iff f = 0$. This property is achieved when restricting $\|\cdot\|_p$ to $L^p$, since if a function $f$ is $0 \ \mu$-a.e., it is in the same equivalence class as $0$, hence $f$ and $0$ are "treated as the same thing". Is this a correct interpretation?

  • 7
    $\begingroup$ Yes, it is correct. $\endgroup$ Apr 17, 2020 at 0:11


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