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Results tagged with measure-concentration
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user 454
13
votes
Accepted
Violating the Lebesgue density theorem
It is a theorem of Besicovitch that measures on $\mathbb R^d$ do satisfy the density theorem.
Fremlin, Measure Theory, Chap. 47
added
Besicovitch, around 1930, extended his density pr …
- 41.1k
4
votes
Accepted
Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$
Let's build a "fat Cantor set". Start with $A_0 = [0,1]$ with measure $\alpha_0=1$.
Then remove a short open interval centered at $1/2$, leaving a set $A_1 \subset A_0$ of measure $\alpha_1 < \alpha_ …
- 41.1k
3
votes
Doob Martingale: Where is the catch?
What if we work it out? All of $X_j$ are equal to $X_0$, in particular measurable with respect to $X_0$. So
$$
\sum_{i=0}^n X_i = (n+1)X_0,
\\
Y_i = (n+1)X_0\qquad\text{for all } i
\\
Y_{i+1}-Y_i = …
- 41.1k