Interchanging the limiting operations How to remember the conditions for interchanging the limiting operations , for example between limits and integrals or integrals and sums or derivation of any order and integrals, i mean every one of these requires more or less different conditions from uniform continuity to normal continuity and the continuity of the derivatives , but i think they all have the same spirit , so what i'm looking for is some 'general statement' to remember them , also if there is an aid like a diagram that may help to easily calling the right statement when it is needed.
 A: The uniform convergence is closely linked to the idea of repeated limits. It's serves as a criterion to validate ownership of Interchange of limiting operations. Riemann integrals and Lebesgue can be seen as limits of Riemann sum and integrals of simple functions, respectively. Soon the question can be answered by the following theorem.
Let $\mathbb{T},\mathbb{X}$ and $\mathbb{Y}$ normed spaces (or metric spaces or topological spaces according to application needs),

Theorem. Let $F:\mathbb{T}\times\mathbb{X}\to\mathbb{Y}$  a function such that: for $x_o\in\mathbb{X}$ and $t_o\in\mathbb{T}$ 
  there are the limits:
1.
  For all $x\in\mathbb{X}$, $\displaystyle\lim_{t\to t_0}\sup_{x\in\mathbb{X}}\left\|F(t,x)-F(x)\right\|=0$ for a given function $F:\mathbb{X}\to\mathbb{Y}$.
2.
  $\displaystyle\lim_{x\to x_0}\left\|F(t,x)-L_t)\right\|=0$ with $L_t$ fix to each$t\in\mathbb{T}$. 
  Then there are the limits
  $$
\lim_{t\to t_0}\| L_t - L\|=0,
\qquad
\lim_{x\to x_0}\lim_{t\to t_0}F(t,x),
\qquad
\lim_{t\to t_0}\lim_{x\to x_0}F(t,x) 
$$
  and the interchange of limit operation 
  $$
\lim_{x\to x_0}\lim_{t\to t_0}F(t,x)
=
\lim_{t\to t_0}\lim_{x\to x_0}F(t,x) 
$$
  holds.

Here, the limit $\displaystyle\lim_{t\to t_0}\sup_{x\in\mathbb{X}}\left\|F(t,x)-F(x)\right\|=0$ is equivalent to uniform convergence. The starting point for the proof is to observe that for any $t_1$ and $t_2$and $ x $ we have the inequalities
\begin{align}
\|L_{t_1}-L_{t_2}\|
\leq
&
\|F(t_1,x)-L_{t_1}\|+\|F(t_2,x)-L_{t_2} \|+\|F(t_1,x)-F(t_2,x)\|,
&
\\
\\
\|F(x)-L\|
\leq 
&
\| F(x)-F(t,x)\|+\|F(t,x)- L_t\|+\|L_t-L  \|
\end{align}
and using the hypotheses 1) and 2) of the theorem of the form properly.The theorem can be illustrated in the following diagram.Above and to the left of the dotted line we have the hypotheses of the theorem. And below and to the right of the dotted line we have the consequences of the theorem.

The notion of interchange of limit operations can be found in classical books of Analysis. For example in the books of Serge Lang. 
This theorem can also be found in a very general version, with the notion of convergence in topological spaces, in Zorich's book:Mathematical Analysis II p. 381).  It reference show in detail all the above mentioned consequences of this theorem. It's a pity we can not visualize them in googlebook's. 
EXEMPLE: If for a sequence of partitions $\{\mathcal{P}_n\}_{n\in\mathbb{N}}$ of interval $[a,b]$ we have 
$$
\lim_{n\to \infty}\sup_{x}\left | \sum_{u_k\in\mathcal{P}_n}f(u_k^*,x)\Delta u_k
- \int_a^bf(u,x)\mbox{d} u\right |=0 
\mbox{ and}
\lim_{x\to x_0}f(u,x)=L_u 
$$ 
then 
$$\lim_{x\to x_0}\int_{a}^{b}f(u,x)\mbox{d} u= \int_{a}^{b}\lim_{x\to x_0}f(u,x)\mbox{d} u $$
To see how this  follows of theorem set


*

*$\mathbb{T}=\mathbb{N}\cup\{\infty\}$ equipped with the topology generated by discrete intervals $[n,m]:=\{n,n+1,\ldots, m-1,m\}$,

*$\mathbb{X}=[a,b]$ and $\mathbb{Y}=\mathbb{R}$ equipped with the topology generated by open intervals,    

*$F:\mathbb{T}\times\mathbb{X}\to\mathbb{Y}$ by 
$$F(n,x)=\sum_{u_k\in\mathcal{P}_n}f(u_k^*,x)\Delta u_k,$$
and apply the above theorem.
