On Thu, Aug 8, 2019, 00:47 Brad Klee <bradklee@gmail.com> wrote:
Andy,
In this case the function doesn't have poles, so you are probably right.
It makes no difference whether the function has poles or not. If the aLebesgie integral exists ( which requires the function to be defined and finite except possibly on a set of measure zero, then changing the function on a set of measure zero does not change the integral.
I still think it's weird that a complete integral can be taken on an incomplete domain, so I'm willing to worry more.
The definition of Lebesgue integration is a fairly fundamental part of analysis. What about it worries you? Andy
Another "problem" is that, after more thought, it seems plausible that the configuration space R would need an odd volume element to match Jim Propp's preferred standard. I think it's still "okay" to start with Cartesian, but would not be surprised if this led to statistical diversity.
Also, nice reading of Keith's algorithm, I was confused by the part about "carving", but now I can see from your perspective that it actually sounds right. Is Keith's algorithm on Mathworld? Maybe we could get EW to add a few more equations?
Please don't blame Fred for listening to me. We are all trying to have fun and learn more, and it doesn't hurt to double or triple check for quality assurance!
Cheers --Brad
On Wed, Aug 7, 2019 at 11:28 PM Andy Latto <andy.latto@pobox.com> wrote:
If what we're interested in is the integral of a function with respect to a particular measure, there is nothing tricky about ignoring measure zero sets. If a set has measure 0, the integral is unchanged by integrating a function that has different, arbitrary, values on that set.
Andy andy.latto@pobox.com
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