Let us define a measure as follows:

$\nu (B) := \pi^{-d/2} \int_B \exp (-\vert x\vert ^2)\mathrm{d}m^d(x)$ where $m^d$ is the $d$-dimensional Lebesgue measure.

Is there a obvious interpretation of this measure?


closed as unclear what you're asking by Nate Eldredge, Dominik, Davide Giraudo, Leucippus, Namaste Dec 7 '16 at 1:32

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    $\begingroup$ This is Gaussian measure with mean zero and variance $1/2$. What do you want to know about it? $\endgroup$ – Nate Eldredge Dec 6 '16 at 15:10
  • $\begingroup$ For the moment that is all I wanted to know, Thanks a lot! Is there also a name for the following measure defined and the sphere $\mathbb{S}^d$: $\nu (A) := \int_A f d \sigma$ where $f$ is an arbitrary fixed continuous function and $\sigma$ is the surface measure? $\endgroup$ – tubmaster Dec 6 '16 at 15:25
  • $\begingroup$ Please ask that as a new question. Asking new questions in comments is bad practice; people can't find them easily. $\endgroup$ – Nate Eldredge Dec 6 '16 at 15:44
  • $\begingroup$ ok, done, sorry for that $\endgroup$ – tubmaster Dec 6 '16 at 15:51

This is Gaussian measure with mean zero and variance $1/2$.

  • $\begingroup$ thanks! I am asking why this measure is inner regular. Do you have a formal proof for that (since we know it is a radon measure)? $\endgroup$ – tubmaster Dec 6 '16 at 15:58

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