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Re: numerical Lebesque integration?
Posted:
Oct 30, 2002 11:59 AM
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Dave Seaman wrote:
>
>>Maybe I am throwing out the baby with the bath water (leaving no
>>interesting examples to exhibit), but if a function is defined on a set
>>S of measure zero in a way that is different from the way the function
>>is defined on all other points in the interval (call the set of such
>>points T) and we happen to stumble upon a point p in S in choosing our
>>random points, we will obviously choose the value of the function at the
>>nearest available member of T rather than the value of the function at
>>the point p, knowing that the value of the function on a set of measure
>>zero doesn't change the Lebesgue integral.
>
>
> We will obviously choose? Why is that obvious?
>
> How would we even know whether any of our sampled points happen to be
> points of continuity for the function, given only the limited data at our
> disposal? All we know about the integrand in general is what we learn by
> probing at a finite number of points.
>
Erng. I said in my post that recognizing that the set of rationals was
a set of zero measure was doing all the heavy lifting. I could just as
well have said that perhaps recognizing that fact was more or less
equivalent to doing the integral analytically. It certainly takes away
whatever is hard or even interesting about Lebesgue integrals in this
example. I could have just responded to your post by saying "Oh, that's
right," but I'm trying to learn more about Lebesgue interals, not to
prove a point.
> And you haven't answered my question about how to evaluate a function at
> an irrational point, using a computer that represents floating point
> numbers with finite precision. If only rational values of x can be used
> in computing (or approximating) f(x), then our entire computation is
> carried out within a set of measure zero.
>
Yeeessss, but the whole point of numerical computation is to find
approximations to functions that can almost never, not even in the most
trivial of cases, be calculated or represented with complete accuracy on
a computer.
What do you do when confronted with integrating a function with a
singularity? If you can, you calculate the contribution of the
singularity analytically and integrate around it or subtract a function
with the same singularity whose integral is known and integrate the
remaining function numerically.
What I proposed is something quite similar for the Dirichlet function.
I know that the Lebesgue integral of a fucntion with the value one one
the rationals and zero everywhere else is zero, so I added that function
to the one you proposed to integrate and integrated the result. The
fact that the function I am left with is Riemann-integrable makes the
example kind of uninteresting.
Classical computers have all kinds of limitations as to what they can
represent and calculate. If we're going to run into a brick wall with
the Lebesgue integral, I'd like to know the size and shape of that brick
wall if I can.
> Besides, you have much too simple a view of what measurable functions are
> like. The Dirichlet function is not at all pathological; it's an example
> of what's called a "simple function" in Lebesgue integration theory.
> That is, it's a function that assumes only finitely many values, each on
> a measurable set.
>
> Most measurable functions are not nearly so well behaved. You can't
> necessarily redefine a measurable function on a null set and make it
> continuous, and you can't necessarily identify a small collection of sets
> on which the function is "defined differently", since the function might
> well be "defined differently" on thousands of measurable sets, each
> scattered throughout the interval of integration, and each having
> positive measure.
>
> How much can we possibly learn about such a function by probing at a
> finite number of rational points?
>
If the challenge is to define an algorithm that could integrate a
function defined by a black box (i.e., you specify an x and the black
box returns a y), then your example of the Dirichlet function settles
the matter definitively, but I would come to the same conclusion about
numerical integration of Riemann-integrable functions if faced with the
same challenge.
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