Recently I've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure-theoretic approach? (For example, why would it be interesting to do probability theory via the expectation operator $\mathbb{E}$ instead of the measure $\mathbb{P}$ on a probability space?)
What are some interesting, yet elementary applications, examples and theorems taking this direction?
I have been interested recently in connections between singular integrals and probability theory. The theory is deep and I can provide a large number of examples of connections of the "functional analytic" approach rather than the "measure theoretic" approach.
First, note that the probability $\mathbb{P}$ represents the probability measure. The symbol $\mathbb{E}$ represents the value of the integral of the set of random variables with respect to the probability measure.
It was Elias Stein who suggested that problems in analysis may be able to be attacked via probabilistic methods as certain variables tend to infinity.
An example is the $L^p-$boundedness of the Riesz transform which is a classic problem in analysis. It has been proven using various techniques, but one method is to consider a probabilistic interpretation of the Riesz transform and then attack the question of $L^p-$boundedness. Given that the Riesz transform is a principal-value integral, $\mathbb{E}$ rather than $\mathbb{P}$ helps with the probabilistic interpretation.
Does that help? I can give more examples in other topics (such as the Dirichlet problem) or go into more detail if you like. Also, I can recommend some good books if you are interested, and provide more examples where expectation is used rather that the probability measure $\mathbb{P}$. Let me know if there is something more specific you wanted to know about.