Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?

Question

Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable function (thought of as a random variable). Then there is a pushforward measure induced on $(S,\mathcal{M})$ (thought of as the probability distribution of $X$), which we could denote $X_*(\mathbf{P}),$ following Wikipedia.

However, I like to imagine that $X$ "knows" that its domain is the whole probability space $(\Omega,\mathcal{F},\mathbf{P}).$ Hence $X$ is "aware of" the probability measure on its domain, and hence we shouldn't have to mention $\mathbf{P}$ in the notation for the pushforward. This coincides better with how I like to write and talk: I will usually just speak of the "probability distribution" of $X$, without mentioning $\mathbf{P}$ at all.

Question. Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$? Something like $\mathrm{distr}(X)$ or $\mathbf{D}(X)$, for example.

Answer 1

Just so that everyone knows what we are talking about here, let me rephrase in more familiar notation.

Suppose $(\Omega, \mathcal{F}, P)$ is a probability space, and $(M, \mathcal{M})$ is a measurable space. If $X : \Omega \to M$ is a random variable (i.e. a $(\mathcal{F}, \mathcal{M})$-measurable function), it induces a pushforward measure on $(M, \mathcal{M})$, which we might denote as $\mu$, defined by $\mu(A) = P(X^{-1}(A))$.

The measure $\mu$ is sometimes called the distribution or law of $X$, and as such I've often seen it denoted by $$\operatorname{Law}(X)$$ or $\operatorname{law}(X)$. See for instance Definition 1.10 of this paper by myself and collaborators, in which we use this notation. We didn't invent it; I don't know who did, but it is quite common in the field, and I think it would be generally understood by probabilists.

For instance, if we had some other measure $\nu$ on $M$ laying around, we could write something like "$\operatorname{Law}(X) \ll \nu$" if it so happened that the one measure was absolutely continuous to the other. In principle you could write something like "$(\operatorname{Law}(X))(B) = 2/3$" but in practice you would simply write "$P(X \in B) = 2/3$" instead. If you plan to use the measure extensively, you should give it a name (e.g. "Let $\mu = \operatorname{Law}(X)$").

As Potato correctly points out, this notation makes it look like the measure only depends on $X$, when of course it also depends on the underlying probability measure $P$. But it is quite standard in probability theory to tacitly assume that everything in sight depends implicitly on the underlying probability space, and thus to suppress it from notation. I guess one could take a categorical interpretation as goblin does, and say that the underlying probability space is "part of" the object $X$, but I don't think most probabilists think that way.

Answer 2

Given a measureable $f:(X,\mathcal{M},\mu)\to(Y,\mathcal{N})$ (a measure space to a measurable space) one can define a measure $\nu=f_{*}\mu$ on $(Y,\mathcal{N})$ by $$ \nu(B)=\mu(f^{-1}(B)). $$ Both $f$ and $\mu$ are necessary to define it, so $f_{*}\mu$ seems like good notation to me.