L^1 function

An $L^1$ function on a measure space $(X,\Sigma,\mu)$ is an equivalence class of measurable functions $f:X\to\mathbb R$ such that

where two functions are identified if they satisfy a.e. equality .

Choosing any representative $f$ of the class, the quantity $\|f\|_1:=\int_X |f|\,d\mu$ is its Lp norm with $p=1$, and the set of all such classes is the Lp space with $p=1$. An $L^1$ function is equivalently a Lebesgue integrable function once a representative is fixed.

Examples:

• On $\mathbb R$ with Lebesgue measure , the function $f(x)=\frac{1}{1+x^2}$ represents an $L^1$ function.
• On the interval $[0,1]$ with Lebesgue measure, the function $f(x)=x^{-1/2}$ represents an $L^1$ function.