Alexey
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 Apr9 comment Decomposition of ball in Banach Tarski paradox and covering a soccer ball I think the two parts are related: with just one soccer ball and Banach-Tarski Theorem one can start soccer ball company. Apr8 comment “Differential” of a measure I agree however that the expression $f(x)dx$ can be generalized outside of the context of differential forms. For example, if $f$ is continuous and $g$ is of bounded variation on $[a,b]$, then $\int_a^b f(x)dg(x)$ is defined (no measure needed either). Apr8 comment “Differential” of a measure If $g$ is continuous and $\mu$ is the Lebesgue measure, then $\int_{\mathbb{R}}g(x)d\mu =\int_{\mathbb{R}}g(x)|dx|$ (no orientation on $\mathbb{R}$ is needed for the second integral). The question remains: what is $g(x)d\mu(x)$? Apr8 comment “Differential” of a measure @epimorphic, $g(x)dx$ is a differential form. To integrate it over $\mathbb{R}$, no measure is needed, only the differential structure (tangent spaces at each point of $\mathbb{R}$, etc.) and the orientation. Apr6 awarded Investor Apr6 comment “Differential” of a measure Just a thought: can the formalism of generalised functions be somehow used to unify $dx$ and $d\mu$? Apr5 comment Negative Zero in the set of real numbers But in any case i think that "equality" did not always mean "identity". Apr5 comment Negative Zero in the set of real numbers Well, judging by the upvotes, i thought that there were some content in the question, and there were something to prove... Apr5 comment Negative Zero in the set of real numbers Can you prove that if two elements are equal, then they are the same element? :) Apr5 comment Negative Zero in the set of real numbers A related question: are $-(-1)$ and $1$ equivalent or equal? Apr4 revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions consider (G,H)-"symmetrization" of H-invariant functions, rewrite, reformat Apr4 comment “Differential” of a measure Moreover, one should ask not only for the meaning of $d\mu(x)$, but for the meaning of $f(x)d\mu(x)$. I would hope for a concise and precise answer, like in the case of $\int_a^b f(x)dx$ (here $f(x)dx$ is a differential form, to be integrated over the oriented interval $[a,b]$). Apr4 comment “Differential” of a measure My intuition suggests me that $d\mu$ must be something like "density" of $\mu$, but i think that in mathematics measure density is a relative notion: only density of one measure with respect to another can be defined. Apr4 comment “Differential” of a measure IMO this does not quite answer the question. Expressions like $dx$, $df$ are just differentials (particular cases of differential forms). What it $d\mu$? Apr3 revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions fix an error in the formulae Apr3 comment “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions Good point, @user86418, but then a coefficient $1/d!$ is also needed. Apr3 revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions suggest "folding" as a term Apr3 revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions add an analogy with adjugate matrix Apr2 revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions consider the case when the Abelian semigroup replaced with an affine space Apr2 revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions note bad choice of the terms like "symmetrization" in the question and the title