Timeline for Notation for calculus with measures?

3 events

when toggle format	what		by	license	comment
Nov 7, 2018 at 14:54	comment	added	Alex Shpilkin		People sometimes write $\int_X f\mu$ instead of $\int_X f\,d\mu$. This yields a good notation for Radon-Nikodym: $\mu = (\mu/\nu)\nu$. To include a coordinate, it would make sense to write $\int_{x\in X} f(x)\mu(x)$ (but people don’t do that). In your example this would give $\int_{x\in X} (f\circ g)(x)\mu(x) = \int_{y\in g(X)} f(y)\mu[g^{-1}(x)]$, properly representing the pushforward measure $g_*\mu(A) = \mu[g^{-1}(A)]$. (People do write $\int_x f(x)\phi(x)\equiv (f,\phi)$ for a distribution $f$ and test function $\phi$, though, so it’s not like the argument $x$ is always taken literally.)
Dec 23, 2016 at 3:14	comment	added	Nate Eldredge		I would probably write the change-of-variables formula as $\int_X f \circ g\,d\mu = \int_Y f\,d(g_*\mu)$.
Dec 23, 2016 at 2:33	history	asked	user13113	CC BY-SA 3.0