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I'm trying to understand a formula of this kind $$ ...=\phi_\sharp \left ( f \mathcal{H}^n \right ) $$ where $\mathcal{H}^n$ is the n-dimensional Hausdorff measure on a measure space $X$, $\phi : X \to Y$ ($Y$ also a measure space) and $\phi_\sharp$ is the pushforward. Also $f:X \to \mathbb{R}$.

Recall that the definition of pushforward requires "$f \mathcal{H}^n$" to be a measure (see for instance GMT of Mattila, definition 1.17).

My question is: how do I actually read the notation "$f \mathcal{H}^n$"? And where can I find references about it?

By definition of pushforward, for $A \subseteq Y$, we have immediately $$ \left [ \phi_\sharp \left ( f \mathcal{H}^n \right ) \right ](A) = \left [ f \mathcal{H}^n \right ] \left ( \phi^{-1}(A) \right ) $$ but that is as far as I go.

PS$1$: I already read this but it didn't help much: Notation for the pushforward measure

PS$2$: since this is a notational question, feel free to change some of the setting if it can help.

Thanks!

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IWillDerive
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1 Answers1

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Usually, if $\mu $ is a measure and $f $ is a function, then $f\mu $ is the measure given by $$ (f\mu)(A) =\int_A f \, d\mu. $$

This is indeed a measure as long as $f \in L^1 (\mu)$ or if $f : X \to [0,\infty] $.

PhoemueX
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