[SciPy-User] [OT] Transform (i.e., Fourier, Laplace, etc.) methods in Prob. & Stats.

josef.pktd@gmai... josef.pktd@gmai...
Thu Nov 26 00:19:38 CST 2009

```On Wed, Nov 25, 2009 at 11:45 PM, David Cournapeau
<david@ar.media.kyoto-u.ac.jp> wrote:
> josef.pktd@gmail.com wrote:
>>
>> Maybe the last statement is wrong, it's too long ago that I
>> struggled with this. Maybe I'm mixing up Lebesgue-integral,
>> Lebesgue-measurable, and measures that are absolutely continuous
>> with respect to Lebesgue-measure.
>>
>
> I am by no mean an expert on this, but I believe you are right. AFAIK,
> contour integrals require to have a piecewise-continuous parametrization
> of your path, and for me, the whole point of Lebesgue integrals is to
> handle cases where the set over which you integrate the function is not
> a (finite) union of intervals.
>
> I don't know if it makes sense to define something "like" contour
> integrals for lebesgue integrals. The fundamental reason why Lebesgue
> integrals work the way they do is because for a function f: E ->F, only
> the properties of F (and how the inversion function maps elements of the
> sigma algebra F) matter. And complex analysis is 'special' because of
> the special structure of E, not F.

I think on the theoretical level I'm right, but from what I read the last few
hours, contour integrals seem to provide a method to actually calculate
the integral, while I haven't seen much practical applications of
Lebesgue integration.

For the simple examples that I tried so far for the inversion of the
characteristic function, I didn't need contour nor Lebesque integrals.
And I hope it stays this way when I get back to this, especially
the special structure of complex numbers.

Josef

>
> David
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