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Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
22
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2
answers
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Gently changing measure
This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{ …
21
votes
1
answer
833
views
Relative null-ness
Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer https://math.stackexchange.com/questions/1444498/is-there-a-categorizaiton-system-for-null …
21
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2
answers
1k
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Antirandom reals
This is a crossposting of https://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it bel …
18
votes
1
answer
766
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Two strengthenings of "strong measure zero"
A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering $X …
11
votes
2
answers
479
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The "strong" measure number
Beyond measure zero we have yet another measure-y notion of smallness: strong measure zero. A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, th …
8
votes
0
answers
537
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A Banach-Tarski game
This is partially inspired by the question https://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written.
A paradoxical family of subsets …
6
votes
1
answer
484
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Probability that a Turing machine will nontrivially reduce a real
For a fixed Turing machine $\Phi_e$, what is the probability that it will reduce a given real to some less complex, yet still non-computable real?
More precisely: It is known that the set of reals wi …
3
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0
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"Nicely" strong measure zero sets
This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\rightarrow\ome …