17 votes

How badly can the Lebesgue differentiation theorem fail?

Metafune has given an example of the limit failing to be $0$ at a particular point - namely for $n > 1$, the function $|x|^{-\alpha}$, with $1 \leq \alpha < n$ has that limit equal to $\infty$ ...
  • 1,674
7 votes

Integration in the surreal numbers

In a comment on Timothy Chow's recent answer that cited a paper a by O. Costin, H. Friedman and myself concerned with integration on the surreals, I noted that the cited paper has now been superseded ...
4 votes

Volume of submanifold as integral of delta-function

One should distinguish between the volume of the submanifold (a number that might be infinite) and the volume form, an exterior differential form $\omega$ of degree $n{-}m$ on the (presumed regular) ...
4 votes

Estimate an improper integral

$\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\De}{\Delta}$This inequality does not hold in general, even if the function $f$ is nondecreasing, which will be assumed henceforth. Indeed,...

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