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Apr
9
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.
Apr
8
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).
Apr
8
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 $g(x)d\mu(x)$?
Apr
8
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.
Apr
6
awarded  Investor
Apr
6
comment “Differential” of a measure
Just a thought: can the formalism of generalised functions be somehow used to unify $dx$ and $d\mu$?
Apr
5
comment Negative Zero in the set of real numbers
But in any case i think that "equality" did not always mean "identity".
Apr
5
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...
Apr
5
comment Negative Zero in the set of real numbers
Can you prove that if two elements are equal, then they are the same element? :)
Apr
5
comment Negative Zero in the set of real numbers
A related question: are $-(-1)$ and $1$ equivalent or equal?
Apr
4
revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions
consider (G,H)-"symmetrization" of H-invariant functions, rewrite, reformat
Apr
4
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 $[a,b]$).
Apr
4
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.
Apr
4
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 $d\mu$?
Apr
3
revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions
fix an error in the formulae
Apr
3
comment “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions
Good point, @user86418, but then a coefficient $1/d!$ is also needed.
Apr
3
revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions
suggest "folding" as a term
Apr
3
revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions
add an analogy with adjugate matrix
Apr
2
revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions
consider the case when the Abelian semigroup replaced with an affine space
Apr
2
revised “Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions
note bad choice of the terms like "symmetrization" in the question and the title