Generalized Sum over Union of Disjoint Index Sets

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Theorem

Let $\struct {G, +}$ be a commutative topological semigroup.

Let $I$ and $J$ be disjoint indexing sets.

Let $K = I \cup J$.

Let $\family{g_k}_{k \mathop \in K}$ be an indexed family of elements of $G$.

Let the generalized sums $\ds \paren{\sum_{i \mathop \in I} g_i}$ and $\ds \paren{\sum_{j \mathop \in J} g_j}$ converge.

Then:

the generalized sum $\ds \sum_{k \mathop \in K} g_k$ converges

and:

$\ds \sum_{k \mathop \in K} g_k = \paren{\sum_{i \mathop \in I} g_i} + \paren{\sum_{j \mathop \in J} g_j}$

Proof

Let $0_G$ be the identity of the semigroup $\struct {G, +}$.

Let $\family{f_k}_{k \mathop \in K}$ be an indexed family of elements of $G$ defined by:

$\forall k \in K : f_k = \begin{cases}

g_k & : k \in I \\ 0_G & : k \in J \end{cases}$

Let $\family{h_k}_{k \mathop \in K}$ be an indexed family of elements of $G$ defined by:

$\forall k \in K : h_k = \begin{cases}

0_G & : k \in I \\ g_k & : k \in J \end{cases}$

From Generalized Sum Restricted to Non-zero Summands:

$\ds \sum_{k \mathop \in K} f_k = \sum_{i \mathop \in I} g_i$

and:

$\ds \sum_{k \mathop \in K} h_k = \sum_{j \mathop \in J} g_j$

From Sum Rule for Convergent Generalized Sums:

$\ds \sum_{k \mathop \in K} \paren{f_k + h_k} = \sum_{k \mathop \in K} f_k + \sum_{k \mathop \in K} h_k = \sum_{i \mathop \in I} g_i + \sum_{j \mathop \in J} g_j$

We have:

\(\ds \forall k \in K: \, \)

\(\ds f_k + h_k\)

\(=\)

\(\ds \begin{cases} g_k + 0_G & : k \in I \\ 0_G + g_k & : k \in J \end{cases}\)

definition of $f$ and $h$

\(\ds \)

\(=\)

\(\ds \begin{cases} g_k & : k \in I \\ g_k & : k \in J \end{cases}\)

Definition of Identity Element

\(\ds \)

\(=\)

\(\ds g_k\)

Hence:

$\ds \sum_{k \mathop \in K} g_k = \paren{\sum_{i \mathop \in I} g_i} + \paren{\sum_{j \mathop \in J} g_j}$

$\blacksquare$