0.2Preliminaries

Updated 9 Sep 2026Chapter (PDF)

This is not a section in Folland and content is drawn from a variety of places, such as Chapters 1-2 of MIRA.

[0.2.1]Definition(Symmetric Difference)#

$E \triangle F = (E \setminus F) \cup (F \setminus E)$ .

The symmetric difference gives all the elements in one set that the other does not contain.

[0.2.2]Definition(Sum over Uncountable Indexing Set)#

Suppose $X$ is an arbitrary set and $f: X \to [0, \infty]$ . Define

\sum_{x \in X} f(x) = \sup\Bigg\{ \sum_{x \in F} f(x): F \subseteq X, F \text{ finite} \Bigg\}.

[0.2.3]Theorem#

Suppose $f: X \to [0, \infty]$ . Let $A = \{x: f(x) > 0\}$ . If $A$ is uncountable, then $\sum_{x \in X} f(x) = \infty$ . If $A$ is countably infinite, then $\sum_{x \in X} f(x) = \sum_{k = 1}^\infty f(g(k))$ , where the right side is an ordinary infinite series and $g: \bZ_+ \to A$ is a bijection.