questions about the Riemann-Lebesgue integral - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:44:24Z http://mathoverflow.net/feeds/question/108694 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108694/questions-about-the-riemann-lebesgue-integral questions about the Riemann-Lebesgue integral Ricky Demer 2012-10-03T07:07:27Z 2012-10-03T07:07:27Z <p>(Assume Countable Choice.)</p> <p>Generalizing arxiv.org/pdf/math/9903103 and www.um.es/beca/papers/BirkhoffBourgain.pdf, <br> the Riemann-Lebesgue integral can be defined as follows: <br><br><br> For $V$ a real $(\text{T}_0)$ topological vector space, $\Omega$ the underlying set of a measure space with measure $\mu$,</p> <p>$\: f : \Omega \to V \:$ a function, and $v$ a vector, $\;\; \displaystyle\int f \: = \: v \;\;$ if and only if</p> <p>for all open subsets $U$ of $V$, if $\:v\in U\:$ then there exists a countable partition $\Gamma\hspace{0.01 in}'$ of $\:\Omega\:$ into non-empty measurable pieces such that [[for all members $x$ of $\Omega$, if $x\hspace{0.02 in}$'s piece has infinite measure then $\:f(x)$ <br> is the zero vector] and [for all countable refinements $\Gamma$ of $\Gamma\hspace{0.01 in}'$, for all choice functions $\: c : \Gamma \to \Omega \:$, $\:$ $\: \displaystyle\sum_{S\in \Gamma}\:(\mu(S) \cdot f(c(S))) \:$ converges unconditionally to an element of $U$]].</p> <p><br></p> <p>Since the definitions provided in both those sources do use refinements and not just $\Gamma\hspace{0.01 in}'$, presumably something would go wrong otherwise. $\:$ I notice that the only way something can go wrong from that is if uniqueness breaks. $\:$ I'm also wondering what difference not requiring the sums to converge might make. <br><br><br> ${}$1. $\:$ What (if anything) is an example where uniqueness would <br> fail if the choice functions and sums used $\Gamma\hspace{0.01 in}'$ instead of $\Gamma$?</p> <p>${}$2. $\:$ Would one actually get something different if one replaced the last line of the definition with: <br> there exists a finite subset $\Gamma_0$ of $\Gamma$ such that for all finite subsets $\Gamma_1$ of $\Gamma$, <br> if $\: \Gamma_0 \subseteq \Gamma_1 \;$ then $\;\;\;\; \displaystyle\sum_{S\in \Gamma_1}\:(\mu(S) \cdot f(c(S))) \;\; \in \;\; U \;\;\;\;$?</p> <p>${}$3. $\:$ Does assuming that $V$ is locally convex change the answers to 1 or 2? <br><br></p>