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Is it possible to define a multiple integral or multiple sums to infinite order ? Something like $\int\int\int\int\cdots$ where there are infinite number of integrals or $\sum\sum\sum\sum\cdots$ . Does infinite valued functions exist (Something like $R^\infty \rightarrow R^n$ ) ?

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You can formalize such things with the concept of product measure. –  Giuseppe Negro Dec 10 '12 at 11:30

3 Answers 3

  1. You can't do simple arithmetic with infinite numbers (e.g. cardinals or ordinals). As basic arithmetic collapses, so does functional analysis. The very fast the $\infty+1=\infty$ should give you pause, even before you get into defining integrals.

  2. There are mathematical theories of infinite numbers and infinite sums. Look at filters, for example. But you have to know what you're trying to achieve.

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Doesn't functions of functions (Functions defined on a space of functions) admit infinite independent variables ? I want to do calculus operations on them (examples are functions defined on the curves in $R^{n}$ or a manifold) even if it's not rigourous but to be able to get correct answers –  naanwa Dec 10 '12 at 11:58
    
I think you might want to take a look at Hilbert Spaces. –  mousomer Dec 11 '12 at 12:26

At this URL we find this item from Zev Chonoles:

BEGIN QUOTE

Let "$\int$" denote $\int_0^x$. We want to find the solution to

$$\int f = f-1.$$

We simply "factor out" $f$, getting $1=\left(1-\int\right)f$. Thus, $f=(1-\int)^{-1}1$.

Using the geometric series,

$$f=\left(1+\int+\iint+\iiint+\cdots\right)1=1+\int_0^x1~dx+\int_0^x\int_0^x1~dx+\cdots$$

Thus,

$$f=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots=e^x,$$

as expected. (This was told to me by Steve Miller)

END QUOTE

(But this does not say how the operation is actually defined.)

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There are no infinite integrals here, only finite integrals of any length. So while the details of every step needs to be handled carefully, the operation $$ \left(1+\int+\iint+\iiint+\cdots\right)1 $$ is defined the usual way, with the only quirk being distribution. –  Arthur Dec 10 '12 at 11:52
    
I think that you define $\int$ as an operator on a Hilbert space of functions ($f$) so it can be defined. you can expand operators in a power series –  naanwa Dec 10 '12 at 11:52

Yes, it is possible to define multiple integrals or sums to infinite order:

here is my definition: for every function $f$ let

$$\int\int\int\cdots \int f:=1$$ and

$$\sum\sum\cdots\sum f:=1.$$

As you can see, I defined those objects.

But OK, I understand that you are looking for some definitions granting some usual properties of the integral. Here is another answer:

it is possible to define integrals of functions between Banach spaces. There are measures on infinite dimensional Banach spaces (for example Gaussian measures) so this might be the concept which is "meaningful" for you. For example you can consider a Gaussian measure on the space of continuous functions $C([0,1])$ induced by a Wiener process and you can calculate integrals with respect to that measure. With some mental gymnastics you can think about those measures and integrals in a way you asked about.

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Well,I meant to do meaningful calculations using them and arrive at correct answers . I'm not a mathematician ,I asked this because it's useful in physics . –  naanwa Dec 10 '12 at 12:09
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My definitions allow for perfectly "meaningful calculations" and they give perfectly "correct answers". And they have one great advantage: they are very easy to calculate. –  Godot Dec 10 '12 at 12:13
    
@naanwa I made a slight update to my answer which might be of more use for you. –  Godot Dec 10 '12 at 12:24
    
Thanks a lot . I want to learn about Gaussian measures ,What do you suggest ? I want to really understand things like how to calculate integrals of functions defined on curves on a manifold ,this leads to integrals over infinite measures. –  naanwa Dec 10 '12 at 12:34
    
@naanwa I am not an expert, but I can suggest you look for "Gaussian measures in Banach spaces" by Kuo. There is also a great book "Stochastic equations in infinite dimensions" by da Prato and Zabczyk - it covers basic things about measures and integration on Banach spaces and quickly goes to applications in stochastic equations in infinite dimensions. –  Godot Dec 10 '12 at 12:44

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