In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) d[M]_s(\omega)\right]$$ where $M$ is a square integrable martingale.
My question is: why did they choose this notation? I know that if we have two measures $Q, R$ then $Q \otimes R$ is the measure defined as $Q \otimes R(A \times B) = Q(A)R(B)$ and then we extend this measure in a unique way to the sigma algebra generated by the sets in the form $A \times B$.
So I guess that there are some similarities between this definition and the previous one, and that $[M]$ can be used to define some sort of measure on $((0, \infty), \mathcal B(0, \infty))$ (where $\mathcal B(0, \infty)$ is the borel sigma algebra on $(0, \infty)$). One guess could be $$[M](A) = \int_0^\infty 1_A(s) d[M]_s$$
but I would not know how to prove that this definition gives raise to the same expression for $E_M$.
So my question are: Does $[M]$ define a measure ? If so, how? And how to show that it gives raise to the same expectation as before?