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 $

2. 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$ )

3. 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

In my current research, I'm confronted with the justification of some facts, and I don't know how to proceed in proving them, so I need to know if there exist some theorems (precisely three theorems) which allow me to do so.

The problem I am investigating is the following: I have an explicit real valued function $f$, DEFINED and CONTINUOUS on each point of $D=]0,1[^4 $. As it is customary to do, let $f(u,v,w,t)$ be the value that this function takes at point of $(u,v,w,t)\in D$: $f$ cannot be defined at the boundary of $D$ and I can't extend its domain of definition $D$ in order to define it on the whole $\overline{D}=[0,1]^4 $, the closure of $D$. I know that $$ f(u,v,w,t)=\sum_{n=0}^{+\infty} f_n(u,v,w,t)\quad\forall (u,v,w,t)\in D $$ where $\{f_n\}_{n\in\Bbb N}$ is a sequence of functions defined and continuous over $D$ which can be extended as continuous function on $\overline{D}$. This makes me think that, for all integers $n$, $$ \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 | f_n(u,v,w,t)| \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t\quad \text{ exists.} $$ And now the questions.

1. What theorem (be it a necessary and sufficient or only a sufficient condition) would allow me to prove the following formula? $$ \begin{split} \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 & f(u,v,w,t)\mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t \\ =&\sum_{n=0}^{+\infty} \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f_n(u,v,w,t)\mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t \end{split} $$

2. What theorem (be it a necessary and sufficient or only a sufficient condition) would allow me to perform any change of the order of integration respect to any of the variables involved, in order to have for example that $$ \begin{split} \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 &f(u,v,w,t)\mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t\\ = & \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t) \mathrm{d}w \mathrm{d}t \mathrm{d}u \mathrm{d}v \\ = & \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f(u,v,w,t) \mathrm{d}v \mathrm{d}u \mathrm{d}t \mathrm{d}w \quad ? \end{split} $$

3. Finally, suppose that one further hypothesis is made over $f$: $f$ depend on a parameter $ a \geq 0$, call it $f_a$ and suppose that $\forall (u,v,w,t) \in ]0,1[^4$ the mapping $a \mapsto f_a(u,v,w,t)$ is $C^{\infty}$ over $ \mathbb{R}_+$: what theorem (again be it a necessary and sufficient or only a sufficient condition) allow me to say that $$ g:a \mapsto \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 f_a(u,v,w,t) \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t \in C^{2}(\Bbb R^+) $$ and $$ g''(a)=\displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 \displaystyle \frac{\mathrm{d}^2}{\mathrm{d}a^2} f_a(u,v,w,t) \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t $$ i.e. would allow me to differentiate twice under the integral symbol?

I know what theorem allowing me to have 1) 2) and 3) in case that $f$ is defined over an interval of $\Bbb{R}$ and so in case a simple integral

Could anybody help me please? Does there exist a freely accessible reference over the internet where I can find such theorems?