Pushforward: from measure theory to differential geometry?

I was wondering if there is a connection between the pushforward from measure theory, and the pushforward from differential geometry?

In measure theory: let $X:(\Omega, \mathcal{A}, \mu) \to \mathbb{R}$ a random variable. The pushforward measure is then defined as $$ \nu = X_\#\mu = \mu \circ P^{-1} $$ and defines a measure on $\mathbb{R}$. Hence, we push the measure $\mu$ to $\mathbb{R}$.

In differential geometry: let $\phi:M \to N$ be a smooth map between two manifolds. The pushforward differential is a map $$ d\phi(x): T_xM \to T_{\phi(x)}N. $$ Hence, we push a tangent vector of $M$ to a tangent vector of $N$.

I understand both definitions. But I am not sure I understand the relation correctly. Is there even a relation? Can we interpret the pushforward differential as a measure? Is the relation given by the Radon-Nikdoym theorem?

Any idea is welcome.

There is a connection because the concept of pushforward is a general notion in mathematics, besides it formalization in category theory or whatever, it is the notion of induction of an structure in a mathematical space from another through a function $f:X\to Y$.

That is: we use the function $f$ to induce some kind of structure on $Y$ from an structure of the same kind in $X$, then we says that we push forward a kind of structure of $X$ on $Y$ using $f$. This is all. Then you can pushforward measures, vector fields (under some conditions), topologies, algebras, etc...