Integrable Function Spaces
Why
We have seen that the integrable functions form a vector space. How about the square integrable functions? And so on.1
Definition
The integrable function spaces are a collection of function spaces, one for each real number $p \geq 1$, for which the $p$th power of the absolute value of the function is integrable.2
Notation
Let $(X, \mathcal{A} , \mu )$ be a measure
space.
Let $p \geq 1$.
We denote the integrable function space
corresponding to $p$ by $\mathcal{L} ^p(X,
\mathcal{A} , \mu , \R )$.
We have defined it by
\[
\mathcal{L} ^p(X, \mathcal{A} , \mu , \R ) = \Set*{
\text{ measurable } f: X \to \R
}{
\int \abs{f}^p d\mu < \infty
}
\]
Let $\C $ denote the set of complex numbers. Similarly for complex-valued functions, we denote the $p$th space by $\mathcal{L} ^p(X, \mathcal{A} , \mu , \C )$.