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While reading articles about statistical inference and SMC algorithms I have come by the following notation (never seen in school) to represent the expectation w.r.t. a measure $\mu$

$$ \mathbb{E}[\phi(X)] = \int\phi(x) \mu(\text{d}x), $$

which I have come to understand means

$$ \int\phi(x) \text{d}\mu(x), \text{ or } \int\phi(x) \mu'(x)\text{d}x. $$

However, when defining the empirical measure $\mu_N(\text{d}x)$

$$ \mu_N(\text{d}x) := \frac{1}{N}\sum^N_{i=1}\delta_{X^i}(\text{d}x), $$

does the notation

$$ \text{d}\delta_{X^i}(x) $$

make any sense? Or can we just use notation

$$ \delta_{X^i}(x)\text{d}x $$ in this case?

Harish Chandra Rajpoot
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Tingiskhan
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1 Answers1

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I don't think I have ever seen the notations $d\delta_{X^i}(x)$ or $\delta_{X^i}(x)dx$. However the former seems correct to me, as $d\mu(x)$ is a common notation. But I don't really like the latter, as it seems to imply that there's a density w.r.t. the Lebesgue measure, which is of course wrong.

If I had to define the empirical measure, I would just write: $$\mu_N=\frac{1}{n}\sum_{i=1}^N\delta_{X^i}$$ which perfectly makes sense and is unambiguous.

When I'm writing an integral I prefer to use the notation $\int f(x)d\mu(x)$, or simply $\int fd\mu$, rather than $\int f(x)\mu(dx)$, but both notations exist and I think it's just a matter of habit

Augustin
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