The $\sharp$ notation in measure theory

Question

What is the meaning of notational subscript character $\sharp$ in measure theory, densities, and samples drawn from distributions?

For example $f_\sharp \pi = \mu$, where $f$ is a function, and $\pi$, $\mu$ are measures.

Answer 1

This likely means that $\mu$ is the pushforward of $\pi$. It means that for any $A$ measurable in the codomain, $\mu(A)=\pi(f^{-1}[A])$.

More generally, from a category-theoretical point of view, if we have a morphism $f$ in a category $\mathcal C$, and some (implicit) covariant functor $\mathcal C\to \mathcal D$, then $f_*$ or $f_\#$ is typically used to denote the corresponding morphism in $\mathcal D$. For contravariant functor, $f^*$ or $f^\#$ is used.

In this case, $\mathcal C$ is the category of measurable spaces with measurable functions and $\mathcal D$ is the category of spaces with all measures, and the functor assigns to each measurable space the set of all measures on it.

A typical example in close proximity of measure theory where $f^*$ is used is when $\mathcal C$ is the category of continuous functions and $\mathcal D$ is the category of Banach spaces, and the contravariant functor is $X\mapsto C(X)$.