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Nov
29
awarded  Popular Question
Nov
24
comment Is there any $F \in \mathscr{F}$ such that $\mu(F)=x$?
@6005. Do you happen to know where Sierpiński is told to have proved the theorem you mention in your answer? While I'm writing this comment, the ref provided on Wiki.en at the url en.wikipedia.org/wiki/Atom_(measure_theory), namely: W. Sierpiński, Sur les fonctions d'ensemble additives et continues, Fund. Math. 3 (1922), No. 1, 240-246 (in French), is wrong, as discussed on MO: mathoverflow.net/questions/222583.
Nov
24
comment Is there any $F \in \mathscr{F}$ such that $\mu(F)=x$?
References for the sigma-additive case are provided on MO: mathoverflow.net/questions/222583. As for the additive case, Sierpiński's main theorem in: Sur les fonctions d'ensemble additives et continues, Fund. Math. 3 (1922), No. 1, 240-246 (in French), provides an affirmative answer when $\Omega = {\bf R}^n$. I don't know what happens in general and haven't checked 6005's answer below (AFAICS, he claims a counterexample to the surjectivity of $\mu$ if $\mu$ isn't more than additive and non-atomic).
Sep
30
revised Does Darboux property imply strong Darboux property?
added 170 characters in body
Sep
30
answered Does Darboux property imply strong Darboux property?
Sep
23
awarded  Autobiographer
Aug
21
comment Does Darboux property imply strong Darboux property?
Incidentally, the assumption that $V$ is infinite is unnecessary and can be weakened to $V$ being nonempty, while it seems to me that the condition ${\sf d}^\ast(V) = 0$ can't be relaxed so easily.
Aug
21
comment Does Darboux property imply strong Darboux property?
Fair enough, there was a serious mistake. I gave it another try: hope I will be luckier. (Btw, you may want to read simple.wikipedia.org/wiki/Exempli_gratia.)
Aug
21
revised Does Darboux property imply strong Darboux property?
Fixed (?)
Aug
21
comment Does Darboux property imply strong Darboux property?
You should tell me who is your $V$: You may find it unbelievable, but I can't read your mind!
Aug
21
revised Does Darboux property imply strong Darboux property?
deleted 90 characters in body
Aug
21
revised Does Darboux property imply strong Darboux property?
deleted 90 characters in body
Aug
21
revised Does Darboux property imply strong Darboux property?
deleted 90 characters in body
Aug
21
revised Does Darboux property imply strong Darboux property?
added 63 characters in body
Aug
21
revised Does Darboux property imply strong Darboux property?
Changed "ultrafilter" to "free ultrafilter" and added a reference
Aug
21
answered Does Darboux property imply strong Darboux property?
Aug
6
revised Non-additive upper logarithmic density: $\ell^\star(X \cup Y) \neq \ell^\star(X)+\ell^\star(Y)$
simplified the exposition and fixed a typo
Aug
6
revised Non-additive upper logarithmic density: $\ell^\star(X \cup Y) \neq \ell^\star(X)+\ell^\star(Y)$
added 9 characters in body
Aug
6
answered Non-additive upper logarithmic density: $\ell^\star(X \cup Y) \neq \ell^\star(X)+\ell^\star(Y)$
Apr
8
answered Non-empty intersection between a compact and an unbounded connected subset of $\mathbb{R}^d$