# Is there a theory that extend real analysis to functions maps into other algebraic structure?

I am studying real analysis now, reading Rudin's book Real and Complex Analysis. One thing confused me is when talking about measurable functions, we assume the function to be, from an abstract space to an abstract space, but when talking about integral, we just integral functions maps into real or complex number field.

Is there any theory about doing integral of functions maps into other adequate algebraic structure, what should I do to learn it? Or any reason only real or complex functions is adequate for integral?

Added: Is there any book recommendation on integral taking value in Banach space?

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Yes, there is a well-developed theory of integrals of Banach space-valued functions; see en.wikipedia.org/wiki/Bochner_integral . –  Qiaochu Yuan May 2 '12 at 14:51
Rudin use measure from a $\sigma$ algebra to $\mathbb{R}$ or $\mathbb{C}$, making you having this sentiment. The notion of measure can be generalized to other spaces. –  Monoide May 2 '12 at 14:55
When I studied measure theory in grad school, we did it with functions taking values in any Banach space; it extends very naturally. –  Arturo Magidin May 2 '12 at 15:08

## 1 Answer

It is easy to extend the notion of integral when either the domain or the codomain (but not necessarily both) are numeric (i.e. $\mathbb{R}$ or $\mathbb{C}$).

An integral is always a limit of sums of the form "value of the function times measure of the set". So, if either your function or your measure are scalar based, you are likely to be able to define an integral (one still needs to take limits, so there are other considerations too).

In a more general case, when both the function and the measure are allowed to be non-scalar, things get more complicated. In particular because this usually involves non-commutativity, and much less of the usual measure theory can be replicated.

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