I wonder the difference between $L^1(\mu\times\nu)$ and $L^1(\mu;L^1(\nu))$, as if partial derivatives can be exchanged with integration in the second spaces in many articles. In Folland's real analysis, Fubini-Tonelli theorem can't be used without the assumptions $L^1(\mu\times\nu)$ or $L^+(X\times Y)$.


Precisely, does the following statement hold:


If
$$\partial\_s f(s,x)=b(s,x)\in L^1([0,T];L^1(B\_R)),$$
where $B\_R\subset \mathbb{R^d}$, then
$$\frac{d}{ds}\int\_{B\_R}f(s,x)\,dx=\int\_{B\_R}b(s,x)\,dx.$$


It is exactly true if $b\in L^1([0,T]\times B\_R)$ by Fubini-Tonelli theorem.



a question about vector valued Banach spaces
There are two parts to your questions and the second hasn’t been touched on so far. Before bringing some suggestions which I hope will be useful, let me add to the information alrady given in the comments in the first part.


The fact that a function on a product $S\times T$ can be regarded as a function on $S$ with values in a space of functions on $T$ or as a member of a suitable tensor product of function spaces on $S$ and $T$ has a long history and attempts to extend this circle of ideas to Banach function spaces goes back at least to Grothendieck.


For simplicity, it is convenient to start with Banach spaces of continuous functions on compacta. In this situation, it is fairly transparent that the space $C(K\times L)$ is naturally identifiable with the space of continuous functions on $K$ with values in $C(L)$ and with a suitable Banach space tensor product of $C(K)$ and $C(L)$.


The case of $L^1$-spaces is rather more delicate, not least because we are dealing with equivalence classes of functions. However, a more or less complete analogy was established about 60 to 70 years ago.


There is, however, one crucial difference between the above cases—the tensor product norms used (the so-called inductive and projective norms respectively).


Now to the second part. You mention conditions on $b$ but none on $f$ for your statement to be true. In fact the only condition you require on the latter is that it be $L^1$ (no smoothness is required!). The reason for this is that such a function defines a distribution and so is differentiable in the distributional sense. The euler-type result you are looking for is then always true—but taking all operations (integration and differentiation) in the sense of distributions.


The usual application of this kind of result is to establish explicit formulae and there it is irrelevant whether the operations are carried out in the classical or in the distributional sense. If, however, you require the former, then one has to add more restrictive conditions and live with much less elegant formulations.


The crucial results that you require can easily be found online, in an elementary text (in english) by J. Sebastião e Silva at the site jss100.campus.ciencias.ulisboa.pt. The text you require is III.1 “Theory of Distributions” which can be found under “publicações,
Textos Didácticos”. The results you require are Th. 8.4.1 (Fubini) and 8.2.4 (Euler).


