Comment Comment by mturmon 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 writeP A = 0.5This 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. parent mturmon 2 years ago
- 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.