Can we exchange the order of integration when one of the integration is over infinite dimensional space? This is related to my previous question here: Optimize over measure on function space .
Let $\mathbf{S}$ be the set of functions from $\mathbb{R}_+$ to $\mathbb{R}_+$ which are weakly increasing. $\delta$ and $z$ are 2 real-valued random variables with some joint distribution. Is the following equality correct? Under what condition?
$\mathbb{E}[ \int_{\mathbf{S}} s(\delta)d\lambda(s)|z]=\int_{\mathbf{S}} \mathbb{E}[s(\delta)|z]d\lambda(s)$
Thanks in advance.
Edit: Say I define a operator $\Gamma(A)=\int_Af(x)d\lambda(f)$, where $A$ is a set of functions. I guess $\Gamma$ is linear since $\Gamma(A_1+A_2)=\Gamma(A_1)+\Gamma(A_2)$. Does it mean the equality is true?
Sounds like you are interested in the Bochner integral (as it generalizes the Lebesgue integral to functions whose values lie in a Banach space) and yes, the Fubini theorem can be generalized for the Bochner integral. In fact, I'm almost sure that the only property which doesn't hold for Bochner that does for Lebesgue is the Radon-Nikodym property. Now if you are referring to functional integration, that is a whole other beast entirely.
Anyways, I am not familiar with your notation and your question is a bit unclear and lacking in context but if you could give more context, I'll do my best to answer the other part of your question.
Edit:
Ok so the Bochner integral is used to integrate infinite dimensional vector valued functions of the form $f:X \to E$ where $E$ is a Banach space and this would look like $f=(f_{1},f_{2},...,f_{n},...)$ and like in finite dimensions, we just integrate each component individually.
First, let us define a measure space $(\Omega, \mathcal{F}, \mu)$. Now, the Bochner integral is defined analogous to the Lebesgue integral (which I assume you are familiar with) so that a function $f:X \to E$ is integrable if there exists a sequence of simple functions $\varphi_{n}: X \to E$ where each simple function is of the form $$ \varphi(x)=\sum_{i=1}^{n} \mathbb{1}_{E_{i}}(x)e_{i}$$
and we have $$ \int_{X}\sum_{i=1}^{n} \mathbb{1}_{E_{i}}(x)e_{i} \, d\mu =\sum^{n}_{i=1} \mu(E_{i})e_{i} $$ where $E_{i} \in \mathcal{F}$ and $e_{i}$ are the values taken in $E$ Then if $f$ is Bochner integrable, we have that $$\lim_{n \to \infty} \varphi_{n} \to f \, \text{a.e.}$$ and $$ \lim_{n \to \infty}\int_{X} \lvert\lvert \varphi_{n}-f \rvert\rvert_{E} \, d\mu=0$$ and we define the Bochner integral as a limit of the Lebesgue integral of simple functions such that $$ \int_{X} f \, d\mu =\lim_{n \to \infty} \int_{X} \varphi_{n} \, d \mu $$ Now, it is important to note how this differs from the Lebesgue integral of real valued functions because you are going to come out with something that looks like $$ \int_{X} f \, d\mu = \left( \int_{X} f_{1} \, d\mu, \int_{X} f_{2} \, d\mu,...,\int_{X} f_{n} \, d\mu,... \right) $$
which is to say that the Bochner integral is vector valued. Now, it has been awhile since I have worked with these concepts and I have a sneaky suspicion that I have left out some details so you may find this more helpful: https://www.math.ucdavis.edu/~hunter/pdes/ch6A.pdf
Now (informally) for Fubini's theorem in this context, say you have a function $f:A \to E$ given by $$f(x,y)=(f_{1}(x,y),f_{2}(x,y),...,f_{n}(x,y),...)$$ where $A \subset \mathbb{R}^{2}$ then you can reverse the order of operation so that $$ \int\int f(x,y) \, dxdy = \int\int f(x,y)\, dydx$$
Now, reading your edit, it seems like you are more interested in functional integration which integrates over spaces of functions and this is a very tricky topic. I'll try to write more about it (or at least what I know) either later on or tomorrow morning but for now, maybe try reading http://arxiv.org/pdf/1009.5966.pdf
http://arxiv.org/pdf/1002.2446.pdf
http://www2.math.uni-wuppertal.de/~ruediger/pages/pdf/bookscover.pdf