Doucet Notation- d in expectation estimate In the book Sequential Monte Carlo Methods in Practice, the beginning of section 1.3.1. I am confused by the notation used 
$$P_N(dx_{0:t}|y_{0:t})=\frac{1}{N}\sum_{i=1}^{N}{\delta_{x_{0:t}^{(i)}}(dx_{0:t})}$$
This is what is referred to as the empirical estimate of the posterior distribution. I would like to know what does $d$ mean inside this equation. Does it simply denote the fact that this equation is an estimate or does it mean as an "instance when the probability density is evaluated" or something else.
 A: *

*Typically $\delta_x(A)$ is the Dirac measure that puts all it's mass on $x$. So it will evaluate to $1$ if $x \in A$, and $0$ otherwise. That is, it's $1$ if $A$ covers $x$, the support. The important thing is that it is a measure, so the arguments to this function are sets. 

*Sometimes in particle filtering people capitalize the subscript, too, to emphasize that the support is random (your random samples/particles). You haven't, but perhaps you meant to. In either event, the subscript will denote the support, so $\delta_{X_{0:t}^i}(dx_{0:t})$ is the measure that evaluates to $1$ if $X^i_{0:t} \in dx_{0:t}$. Or in other words, if your $i$th particle is in the set that is chosen. 

*He is writing $dx_{0:t}$ instead of $x_{0:t}$ because he wants to be general and not talk about continuous or discrete random variables specifically. Also, he wants to emphasize what is the argument to the measure, and what's the support. So the Lebesgue integral, which has the same definition for both continuous and discrete r.v.s, of any function can be taken with respect to this empirical/approximate measure:
\begin{align*}
E_{P_N(dx_{0:t}|y_{0:t})}[h(X_{0:t})] &= \int h(x_{0:t}) p_N(dx_{0:t}) \\
&= \frac{1}{N}\sum_{i=1}^N h(X^i_{0:t}).
\end{align*}
If you have a page number you want me to refer to, or if I was wrong about $dx_{0:t}$ not having a superscript, feel free to comment below. I have the book in my office I think. 
