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Results tagged with integration
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user 51695
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
3
votes
Accepted
Interchange of integration and supremum
I don't think your left hand side is well defined for the class of $u$ you are considering, I can change each $u(.,t)$ to a large value on the zero-set $S_{|t|}$, which will result in the supremum pic …
- 2,504
3
votes
Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
atlas $(\Phi_i:U_i \to M)_{i\in I}$, a matching partition of unity $(\psi_i: M \to [0,1])_{i \in I}$, maybe an additional fixed weight $g:M \to (0,\infty)$, or even $g:M \to \mathbb{R}$ and set your integration … operator to be
$$I_{\text{naive}}(f) := \sum_{i\in I} \int_{U_i} f(\Phi_i(x)) \psi_i(\Phi_i(x)) g(\Phi_i(x)) dx $$
where you can now use Riemann, Lebesgue or whatever type of integration on $\mathbb{R …
- 2,504