In my research i'm confronted to some justification that i don't know them so i need some theorem (3 theorem exactly)
I have an explicit function DEFINED and CONTINUED in each point of $D=]0,1[^4 $ and taking values over the real number, let $f(u,v,w,t)$ the value that this function take in a point of $D$. The function $f$ have problems of definition at the board of $D$ and i can't extend $D$
On each point $(u,v,w,t)$, i have $f(u,v,w,t)=\displaystyle \sum_{n=0}^{+\infty} f_n(u,v,w,t)$ where $f_n$ are defined and continued over $D$ and can be definied and continued (over a more big domain )=$[0,1]^4$ (and so i think that $ \forall n $ integer $\displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 | f_n(u,v,w,t)| du dv dw dt $ existe)
1)what good theorem (or sufficient condition) allow me make the following
$ \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t)du dv dw dt= \displaystyle \sum_{n=0}^{+\infty} \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f_n(u,v,w,t)du dv dw dt $
What good theorem ( or sufficient condition ) allow me make to change any order for integration \ ( for exemple to have $ \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t)du \; dv\; dw \; dt \; = \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t)dw \; dt \; du \; dv= \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t)dv \; du \; dt \; dw$ )
suppose at last further more the hypothesis made over $f$, that $f$ depend on a parametre $ a \geq 0$ i'll call it $f_a$ and suppose that $\forall (u,v,w,t) \in ]0,1[^4$ $a \rightarrow f_a(u,v,w,t)$ is $C^{+\infty}$ over $ \mathbb{R}_+$
what theorem or sufficient condition allow me to say that $g:a \rightarrow \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f_a(u,v,w,t)$ is two times derivable over $ \mathbb{R}_+$ and$ g''(a)=\displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 \displaystyle \frac{d^2f}{da} f_a(u,v,w,t)du dv dw dt$
i know the theorem allowing me to have 1) 2) and 3) in case that $f$ is defined over an interval of $\mathbb{R}$ and so in case a simple integral
Any help please? is there an easy good chapter over internet where can i find such theorem