Extended L1 Functions

Last updated Nov 1, 2022

Let $(X, \mathcal{M}, \mu)$ be a Measure Space. $$\bar{L^{1}}(X) := {f: X \to \bar{\mathbb{R}} : f \text{ is } \mathcal{M}\text{-measureable}, \int\limits_{X} f^{+} d \mu < \infty \text{ or } \int\limits_{X} f^{-} d \mu < \infty}$$ If $f \in \bar{L^{1}}$ and $\int\limits_{X} |f| d \mu = \infty$, then

1. if $\int\limits_{X} f^{+} d \mu = \infty$, we define $\int\limits_{X} f d\mu = \infty$
2. if $\int\limits_{X} f^{-} d \mu = \infty$, we define $\int\limits_{X} f d \mu = -\infty$

• L-Plus Functions
• L1 Integrable Functions