I guess if P(w) has a derivative we can write it that way. I got the expression from a textbook by Patrick Billingsley. Generally, when P(w) is not differentiable (as shown in the example), we can not write the expression in that form.
P(dw) is like a distribution fuction... may be. I am confused about P(dw), is it probability of dw? Then what is dw? Following the above example, say, we have f(w)=1 for w=1 and 0 otherwise. What is the meaning of dw here and hence value of P(dw) at w=1? I guess the above integral would give...
What does it mean by
\int f(w) P (dw)
I don't really understand P (dw) here. Does it mean P (x: x \in B(x, \delta)) for infinitely small \delta ?
For example, with P(x)=1/10 for x=1, 2, ..., 10 . How can we interpret this in term of the above integral
Thanks...