Understanding SMC as approximations to a sequence of measures

I am used to thinking about Sequential Monte Carlo (SMC) methods as a method that discretely approximates a sequence of probability distributions. For example, in a state-space framework, SMC can yield an approximation to the state distribution at each iteration. This is also how it is explained in much applied work.

More formal sources refer to SMC approximations as a sequence of measures. I am vaguely familiar with the idea of measures and sigma algebras, but I don't fully understand their role in SMC. For example, this article by Jasra and Del Moral contains the passage:

A process is written $\{X_t\}_{t\in[0,T]}$. A measurable space is denoted $(E,\mathscr{E})$. Given a sequence of spaces $E_0,\dots,E_n$ (resp. $\sigma$-algebras $\mathscr{E}_0,\dots,\mathscr{E}_n$) the product space is written as $E_{[0,n]}$ (resp. product $\sigma$-algebra $\mathscr{E}_{[0,n]}$). For a probability $\pi$ and $\pi$-integrable function $h$, the notation $\pi(h):=\int h(x)\pi(x)\mathrm{d}x$ is sometimes used. The Dirac measure on $\{x\}$ is written $\delta_x(\mathrm{d}x')$. Probability densities are often assigned a standard notation $p$.

The following things are not clear to me:

• What is the relationship between the process $\{X_t\}_{t\in[0,T]}$ and the sequence of spaces $\{E_i\}_{i\in[0,n]}$ (or $\sigma$-algebras)
• What is the difference between saying that SMC approximates a sequence of distributions and saying that it approximates a sequence of measures. Which one is formally more correct?
• The Dirac measure is also often used without the differential operator. What is the difference in interpretation and which is more appropriate?

Any help is much appreciated. Note that I am posting this question on stats.stackexchange.com because I want to understand the concepts in order to do empirical work and a purely formal answer may not help me tie the concepts back to their applications.

1. They are just different ways of talking about the same thing. Recall that a random variable is a function that's "measurable." For any $$t$$, $$X_t$$ is just a label for the random variable's output, while $$E_t$$ could be the set of inputs for that random variable. The sigma field for one of these random variables, $$\mathcal{E}_i$$, is a bunch of collections of the inputs; this is useful for defining probabilities.