# About a multiple integral [closed]

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?

## closed as off-topic by abx, Ben McKay, user44191, Sean Lawton, Jan-Christoph Schlage-PuchtaJun 12 at 10:56

This question appears to be off-topic. The users who voted to close gave these specific reasons:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – abx, Sean Lawton, Jan-Christoph Schlage-Puchta
• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Ben McKay, user44191
If this question can be reworded to fit the rules in the help center, please edit the question.

The condition $$\sum_{n=1}^\infty \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 | f_n(u,v,w,t)| \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t < +\infty$$ will let you do 1,2. Alternatively, $$f_n(u,v,w,t)\ge 0$$ will also let you do 1,2 wit the proviso that you have to allow value $$+\infty$$ for both sides.
• thanks mister Gerarld for your help. for having 1) and 2), do i need , further more the condtion you wrote , $\displaystyle \sum_{n \geq 0} \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 | f_n(u,v,w,t)| \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t$ converge? thanks – mamiladi Jun 11 at 1:32
• @mamiladi ... you are right, I forgot the $\sum$ in there. I added it. – Gerald Edgar Jun 11 at 11:57