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Results tagged with real-analysis
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user 61785
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
13
votes
Accepted
Is there a measure on $[0,1]$ that is 0 on meagre sets and 1 on co-meagre sets
The answer is no. Assume that such a measure $\mu$ exists.
First, since every singleton in $[0,1]$ is closed with empty interior, $\mu(\{x\}) = 0$ for all $x \in [0,1]$. Write $B_{x,\epsilon}$ for th …
- 3,768
0
votes
Density character of a metric space is an Ulam number
An Ulam number is some authors' term for what is called "a cardinal smaller than the first real-valued measurable cardinal" by set theorists in general, or "a measure-free cardinal" by D. H. Fremlin ( …
- 3,768
3
votes
Non-separable metric probability space
Iosif Pinelis has given an answer to question 1 and partial answers to 2 and 3. Since he advised me to turn my comments into an answer, here it is.
I will deal with the case where the axiom of choice …
- 3,768
4
votes
Accepted
Baire category theorem for uncountable unions
The hyperstonean case can be dealt with using a result from Fremlin's Measure Theory. For every hyperstonean space $X$, we can find a semi-finite measure $\mu$ defined on the sets with the Baire prope …
- 3,768