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In page 19 of the article "Equivariant cohomology with generalized coefficients" the authors say:

Let $V $ be an n-dimensional real vector space, let $v'$ be a non-zero element in $ \wedge^n V^*$. We denote by $|v'| ^{-1}\delta_V$ the element of $C^{-\infty}(V)$ defined by $$\int_V |v'| ^{-1} \delta_V(X) \phi(X)dX = \phi(0)$$ for any test function $\phi$ on $V$, where $dX$ is the euclidean density on $V$ determined by $v'$.

My question is what is the definition of the Euclidean density $dX$ (determined by $v'$)?

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    $\begingroup$ The translation invariant density, i.e. the Lebesgue measure of Euclidean space. See J.C. Alvarez Paiva and E. Fernandes, Gelfand Transforms and Crofton Formulas for an introduction to densities. We can say that $v'$ is a translation invariant differential form, so we integrate using it: $dX=v'$. $\endgroup$
    – Ben McKay
    Commented Sep 8, 2021 at 12:51
  • $\begingroup$ Thanks a lot @Ben McKay for recommending this book. $\endgroup$
    – Mira
    Commented Sep 8, 2021 at 13:20

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