Integral of vector valued function

[jostpuur]

Let $(X,d\mu)$ be measure space, $Y$ be a vector space, and $f:X\to Y$ some function. If $A\subset X$ is some measurable set, how could we define (or try to define) the integral

$$\int\limits_A f(x) d\mu(x)$$

?

If the $Y$ has a basis $e_1,e_2,e_3,\ldots$, I can define the integral as a vector in this basis by integrating the function component-wisely:

$$\int\limits_A f(x)d\mu(x) = \Big( \int\limits_A f_1(x)d\mu(x),\; \int\limits_A f_2(x)d\mu(x),\; \ldots \Big) \;\in\; Y$$

But is it certain that the integral will be independent of the basis?

I see that the integral will be independent of the basis if $\textrm{dim}(Y)<\infty$, because the transformations of the representations of the function and integral are going to be represented by the same matrices, and because the integral commutes with finite sums.

But integrals don't always commute with infinite sums.

So if we have no preferred basis for $Y$, and if $\textrm{dim}(Y)=\infty$, what can we do?

 

[mathman]

I believe you would need some restriction on f(x).

 

[rrogers]

Might your problem might be covered in "Hilbert space" theory?