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Results tagged with measure-theory
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user 5963
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
16
votes
Accepted
Is Conway's base-13 function measurable?
Call the support set $S$. The answer is yes it is Lebesgue measurable and no, it has zero measure. It is even Borel measurable, which would take a tiny bit more effort to prove.
Note that $S$ is in …
- 8,501
10
votes
The Fundamental Theorem of Calculus in Lebesgue Theory
As I recall Chapter 7 of Rudin's Real and Complex Analysis has a good presentation of the Fundamental Theorem of Calculus in the context of Lebesgue integration.
- 8,501
7
votes
Symmetries of probability distributions
Maps such as $\eta$ and $\xi$ are called measure-preserving and are studied in ergodic theory. In particular ergodic theory views these as dynamical systems, because the maps can be iterated. One th …
- 8,501
5
votes
Accepted
Measurability of sets of pairs
Without regularity at least, such pathologies can happen. In fact, taking products is not necessary to get them. It can happen that with $A\subset X$ we have $\nu_1(A)\neq \nu_2(A)$. Taking the pro …
- 8,501
5
votes
Accepted
Transfer independance from $\mathbb{N}$ to $\mathbb{N}^2$
If the sequence $X_1,X_2,\ldots$ has finite range, then this is impossible except in the trivial way mentioned by Emil.
The "input" random variables (e.g. $X_1$) all have equal and finite entropy $a$ …
- 8,501
3
votes
Extreme couplings
The finite and infinite cases are actually quite different for this problem. In the simpler version of the finite case where $X=Y$ and $\mu=\nu$, you can show that the extreme points are introduced b …
- 8,501
1
vote
Accepted
identically distributed random variables and measure-preserving transformations
No. Take for example the case where $\Omega = [0,1]$ with Lebesgue measure, $X$ is the identity, and $Y = 2X\mod 1$. Then you ask for $T = X\circ T = Y$ almost surely, but changing $Y$ on a set of m …
- 8,501