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Results tagged with integration
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user 165275
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
Is there a standard way of defining the integral of an extended real function with respect t...
Once the integral over measurable bounded functions is defined, a “standard” way is this: For a nonnegative (measurable) function $f$ one takes the supremum of the integral over all bounded measurable …
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3
votes
Accepted
Is this operator continuous?
I do not have a counterexample, but a strong feeling that the conjecture is false, based on the following positive proof.
If you require slightly more, namely Lebesgue integrability of $t\mapsto f(t,x …
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1
vote
Conditions for continuity of an integral functional
Since $\nu$ does not appear anywhere else in the question, I suppose that $L^1(X)=L^1(\nu)$.
In order that the functional be defined, one should then assume (probably without loss of generality) that …
- 1,869
1
vote
Accepted
A "uniform continuity" type condition on a Hammerstein integral equation
This does certainly not follow from your other hypotheses, as what you want to conclude is not much weaker than the equi-integrability of $\{K(t,\cdot):t\in I\}$ (sometimes also called absolute contin …
- 1,869
1
vote
Accepted
Simple example of Hammerstein integral equation
It is a question what is reasonable and whether you can conclude something interesting from it: If, for instance, the integral operator with kernel $\lvert K\rvert$ acts in $L_\infty$ and $f$ grows su …
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1
vote
Bochner integral over convex sets lies in the convex set?
I had the same problem many years ago (if I remember correctly), and the answer was negative. I think that I had found a counterexample in the monograph of Diestel and Uhl.
If $E$ has finite dimension …
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