Integrable Function Space

Why

The integrable functions are a vector space.

Definition

The interable function space corresponding to a measure space is the set of real-valued functions which are integrable with respect to the measure. The term space is appropriate because this set is a real vector space. If we scale an integrable function, it remains integrable. If we add two integrable functions, the sum is integrable. Thus, a linear combination of integrable functions is integrable. The zero function is the zero element of the vector space.

The open question is: what elements of our geometric intuition can we bring to a space of functions. Do functions have a size? Are certain functions near each other?

Notation

Let $(X, \mathcal{A} , \mu )$ be a measure space.

We denote set the real-valued integrable functions on $X$ by $\mathcal{I} (X, \mathcal{A} , \mu , \R )$, read aloud as “the real integrable functions on the measure space X script A mu.” We denote set the complex-valued integrable functions on $X$ by $\mathcal{I} (X, \mathcal{A} , \mu , \C )$, read aloud as “the complex integrable functions on the measure space X script A mu.” When the field is irrelevant, we denote them by $\mathcal{I} (X, \mathcal{A} , \mu )$, read aloud as “integrable functions on the measure space X script A mu.” The $\mathcal{I} $ is a mnemonic for “integrable.”