Definitely. Taking this to its logical end leads to a beautiful concise notation that I learned from David Pollard:

- You literally identify sets with their indicators: they are the same.

- You identify the “P” operator as expectation (integration) with respect to the underlying measure, and next…

- You note that integration is linear, so you use linear operator notation everywhere you’d use P.

So if “A” is a set, you just write

P A = 0.5

This is equivalent to:

P A = P 1[ω ∈ A] = ∫ 1[ω ∈ A] dP(ω)

in Lebesgue notation.

There’s an example on page 2 of http://www.stat.yale.edu/~pollard/Courses/241.fall2014/notes... although he’s not using measure theory there.

It can be really clean and terse especially when doing bounds for random variables.