Riemann-Stieltjes Integral

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>>8470566
>to_what_import.bmp
you could at least give the definition. afair it's the integral rudin uses in his book

also from what I remember
>riemann integral
sum up rectangles under graph.
chop up domain (blue)

>lebesgue integral
same thing, but chop up codomain (red)

where does the riemann-stieltjes integral fit in between these two?

>>8470623
Well to give you a tl;dr of the definition:

You have two functions [math]f [/math] and [math] \alpha [/math] from [math] \mathbb{R} \supseteq D \to \mathbb{R} [/math], then the Riemann-Stieltjes integral is defined as

[eqn] \int_A f \mathrm{d} \alpha = \sum_{j = 0}^{n} f(t_j) (\alpha(x_{j+1}) - \alpha(x_j)) [/eqn]

With each [math] t_j \in [x_j, x_{j+1}] [/math]. This is under the condition that [math] \alpha [/math] is of "bounded variation", meaning:

[eqn] \sum_{j = 0}^{n} (\alpha(x_{j+1}) - \alpha(x_j))^{2} < \infty [/eqn]

Whenever [math] \mathrm{mesh} \mathcal{P} \to 0[/math], with [math] \mathcal{P} [/math] the partition of the interval.

>>8470566
https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration
This shit is useful, Reiman version is a special case (pretty much). The reason you have seen it in stochastic calculus is because with it you can more or less introduce Ito integration without talking about measure theory.

>>8470623
Do you not have google? Either way it is the RIeman integral if you only have countable many discontinuities (defined in the obvious fashion).

You have Euclid's Geometry and its axioms that create a universe you can deduce. This is easy because a lot of what you do in that universe you do in the one you made up to pick your nose and walk across the street.

You have Riemann's geometry and its axioms that are different, but also deduce a universe.

You can compare the two with transforms as long as you are careful about what you are comparing AND you don't take all the conclusions from one and put it in the other..

That is math in a nutshell. Deductive universes are not always transformable for all conclusions, and so we have to be careful..

Integration as the area under a curve looks like, regardless of how you do it, you should come up with the same answer, but but but!

The space underneath that is deduced by the rules is not the same space from one integration technique to the other...

So we need to be careful as to how we define integration.
Riemann-Stieltjes integral is one way of defining integration that the space underneath has certain properties that allow us to make one set of deductions while Lebesgue integration is a different set of deductions and a different space..

As we come up with different deductive universes that we may want to connect, we have to come up with the tools to connect them that preserves the deductive connections that made them in the first place.

Try integrating p-adic measure sometime for a simple concept like a transitivity law.
Or try any kind of integration of a macro supply and demand space on rent data, and watch conservation laws evaporate.

>>8470623
The standard integral (both lebesque and riemann) arise from the idea of finding the area below a function that we can easily calculate: a constant function. Then you prove that the sum of these rectangles converge to the area below a function as your mesh tends to zero.

Riemann-Stieltjes is similar: as this anon >>8470647
pointed out it is basically the same but now you apply the function [math]\alpha[\math] to the extrema of each interval.

Ito integral does the same with a stochastic process, with a coarse mesh you make a new stochastic process that is constant in the subintervals, and since you know how to integrate that constant with respect to a process, you make an integral. Repeat with more refined meshes as the constructed process approaches the process you were trying to integrate and you get the integral.

I could be mistaken since I first learned this literally 1 week ago.

>>8470699
Well the way you say it sounds a little simpler than it is. The ito integral is the integral of a stochastic process (in t), let's say H, also w.r.t. a stochstic process (in t), "dX", where dX is constructed from a stochatic process X (as difference of stochastic processes, similar to \alpha in >>8470647) and such that a limt n to infinity makes dX a stochstic process with zero expectation. The grid is in t.

>>8470566
No 'purpose'... it's a slightly generalized version of the Riemann integral. Lebesgue is better but harder to work with.

>>8470663
W... what?

>>8470734
yes I missed a few details

>>8470734
I thought the only difference between the Riemann-Stieltjes and Ito is that for every interval, you take the leftmost point as height of the step function, whereas in a Riemann-Stieltjes you take any random point in the interval.