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We are looking at the measure space $(\mathbb{R}^d, \mathcal{P}(\mathbb{R}^d),\delta_p)$, for all $A\subset \mathbb{R}^d$:

$\delta_p(A):=\begin{cases} 1, & \text{if } p \in A \\ 0, & \text{otherwise} \end{cases}$.

Let $A \subset \mathbb{R}^d$ and $f:\mathbb{R}^d \rightarrow \overline {\mathbb {R}}$ arbitrary. Show that $f$ is measurable and calculate $\int_A f \space d \delta_p$.

This is probably not that difficult but I don't know how to do it. :(

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1 Answer 1

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$\delta_p$ puts all the mass into the point $p$, so the integral is

$$ \int_{A}fd\delta_p= \begin{cases} f(p) & \text{if}~ p\in A\\ 0 & \text{otherwise} \end{cases}. $$

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  • $\begingroup$ How do you show this rigorously? $\endgroup$
    – Dispersion
    Nov 30, 2020 at 1:15
  • $\begingroup$ A straight answer without any explanation is not very helpful... $\endgroup$ Dec 26, 2020 at 0:22

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