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Apr
7
asked In how many steps is it necessary to construct the Borel $\sigma$-algebra?
Apr
7
comment Function coincides with a function of bounded variation almost everywhere
@user104254 I saw them in Stein's book, but apparently not familiar. It seems that the condition is somewhat different (though the key idea is Dirac $\delta$-shaped functions) and it claims that the convolutions converges pointwisely to the original function at Lebesgue points.
Apr
7
comment Function coincides with a function of bounded variation almost everywhere
@Thomas Edited. The network got stuck, and I even LOST a post. Now I need to repost it.
Apr
7
revised Function coincides with a function of bounded variation almost everywhere
added 71 characters in body
Apr
7
asked Function coincides with a function of bounded variation almost everywhere
Apr
7
comment Outer measure and Caratheodory's criterion
Take a similar example, the topological spaces could be defined through neighborhoods, see wiki‌​. One can omit the fourth axiom to define a topology, but one cannot reproduce the same system of neighborhoods if the fourth axiom is omitted.
Apr
7
comment Outer measure and Caratheodory's criterion
That's not the one I'm looking for. I'm asking whether an outer-measure in the sense of Caratheodory satisfies the infimum property.
Apr
6
answered On construction of a Hamel basis which is also a Bernstein set
Apr
6
asked Outer measure and Caratheodory's criterion
Apr
6
comment Is Cantor-Bendixson theorem right for a general second countable space?
Even if it's Hausdorff, it's not necessarily metrizable, right?
Apr
6
asked Is Cantor-Bendixson theorem right for a general second countable space?
Apr
6
asked On construction of a Hamel basis which is also a Bernstein set
Apr
6
accepted Determining the measure (zero) from the measure (zero) of the intersections with translations
Apr
6
comment Determining the measure (zero) from the measure (zero) of the intersections with translations
Quite interesting and informative. Since $\mathcal C+\mathcal C=[0,2]$ where $\mathcal C$ is the triadic Cantor set, we can choose a Hamel basis as a subset of $\mathcal C$, right?
Apr
6
accepted Connectivity and Euler characteristic for surfaces
Apr
6
asked The principal curvatures of a surface of revolution
Apr
6
asked Determining the measure (zero) from the measure (zero) of the intersections with translations
Mar
25
awarded  Popular Question
Mar
13
comment Notation in derivation theorem
@GitGud Another reference: en.wikipedia.org/wiki/Einstein_notation
Mar
13
comment Notation in derivation theorem
@GitGud The notation $x^k$ for coordinates isn't that bad. For example, $x=\sum x^ke_k$ represents an element of $\mathbb R^n$, while $f=\sum f_ke^k$ represents a linear functional on $\mathbb R^n$. I saw that in other books, say, Spivak's Calculus on Manifolds.