Sequences and Series/Multiple limits

Theorem (interchanging summation and integration):

Let $(\Omega ,{\mathcal {F}},\mu )$ be a measure space, and let $(f_{n})_{n\in \mathbb {N} }$ be a sequence of functions from $\Omega$ to $\mathbb {K} ^{d}$, where $\mathbb {K} =\mathbb {R}$ or $\mathbb {C}$. If either of the two expressions

$\int _{\Omega }\sum _{n=1}^{\infty }\|f_{n}(\omega )\|_{\infty }\mu (d\omega )$ or $\sum _{n=1}^{\infty }\int _{\Omega }\|f_{n}(\omega )\|_{\infty }\mu (d\omega )$

converges, so does the other, and we have

$\int _{\Omega }\sum _{n=1}^{\infty }f_{n}(\omega )\mu (d\omega )=\sum _{n=1}^{\infty }\int _{\Omega }f_{n}(\omega )\mu (d\omega )$.

Proof: Regarding the summation as integration over $\mathbb {N}$ with σ-algebra $2^{\mathbb {N} }$ and counting measure, this theorem is an immediate consequence of Fubini's theorem, given that integration and summation are defined pointwise. $\Box$

Theorem (interchanging summation and real differentiation):

Let $(f_{n})_{n\in \mathbb {N} }$ be a sequence of continuously differentiable functions from an open subset $U$ of $\mathbb {R} ^{d}$ to $\mathbb {R} ^{k}$. Suppose that both

$\sum _{n=1}^{\infty }\|f_{n}(x)\|_{\infty }$ and $\sum _{n=1}^{\infty }\|Df_{n}(x)\|_{\infty }$

converge for all $x\in U$, and that for all $x\in U$ there exists $\delta >0$ and a sequence $(a_{n})_{n\in \mathbb {N} }$ in $\mathbb {R}$ such that

$\sum _{n=1}^{\infty }a_{n}<\infty$ and $\forall n\in \mathbb {N} :\forall y\in {\overline {B_{\delta }(x)}}:a_{n}\geq |Df_{n}(y)|$.

Then

$D\sum _{n=1}^{\infty }f_{n}(x)=\sum _{n=1}^{\infty }Df_{n}(x)$

for all $x\in U$.

Proof: $\Box$