Pushing a Haar measure by a linear map #
We show that the push-forward of an additive Haar measure in a vector space under a surjective
linear map is proportional to the Haar measure on the target space,
We deduce disintegration properties of the Haar measure: to check that a property is true ae,
it suffices to check that it is true ae along all translates of a given vector subspace.
TODO: this holds more generally in any locally compact group, see [Fremlin, Measure Theory (volume 4, 443Q)][fremlin_vol4]
The image of an additive Haar measure under a surjective linear map is proportional to a given additive Haar measure. The proportionality factor will be infinite if the linear map has a nontrivial kernel.
The image of an additive Haar measure under a surjective linear map is proportional to a given additive Haar measure, with a positive (but maybe infinite) factor.
Given a surjective linear map
L, it is equivalent to require a property almost everywhere
in the source or the target spaces of
L, with respect to additive Haar measures there.
Given a linear map
L : E → F, a property holds almost everywhere in
F if and only if,
almost everywhere in
F, it holds almost everywhere along the subspace spanned by the
L. This is an instance of a disintegration argument for additive Haar measures.
To check that a property holds almost everywhere with respect to an additive Haar measure, it suffices to check it almost everywhere along all translates of a given vector subspace. This is an instance of a disintegration argument for additive Haar measures.