Definition of space $L^2(\mu)$ where $\mu$ is a Borel probability measure on $\mathbb R$.

Question

Let $\mu$ be a Borel probability measure on $\mathbb R$ with compact support. Consider the space $L^2(\mu)$. It is the first time that I meet this space (usually I have $L^2(\mathbb R)$). Is it still a Banach space? Is it still a Hilbert space? By what norm and scalar product definitions?

Thanks in advance.

Any $L^2$ space is a Hilbert space (and thus a Banach space). — Math1000, Aug 21 '15 at 18:10

score 2 · Accepted Answer · answered Aug 21 '15 at 18:04

2

All the definitions that you know from Lebesgue integration on $\mathbb{R}$ carry over, with integration against the Lebesgue measure replaced by integration against $\mu$. Most of the core theorems go through as well. This is covered in any decent measure-theoretic real analysis text, e.g. Royden and Fitzpatrick.

answered Aug 21 '15 at 18:04

Ian

101,645

1

With $(f,g)=\int_{\mathbb R} f (t) \overline{g}(t) d\mu(t)$ as scalar product for $L^2(\mu)$? – Lely Aug 21 '15 at 18:11
2

@Lely Correct. ${}$ – Ian Aug 21 '15 at 18:24

Definition of space $L^2(\mu)$ where $\mu$ is a Borel probability measure on $\mathbb R$.

1 Answers1