L^p space

An $L^p$ space (for $1\le p\le\infty$) associated to a measure space $(X,\Sigma,\mu)$ is the collection

where $\|f\|_p$ is the $L^p$ norm and $f\sim g$ means $f=g$ almost everywhere (i.e. a.e. equality ).

The quotient by a.e. equality ensures that the $L^p$ norm is well-defined on $L^p(X,\Sigma,\mu)$. The cases $p=1$ and $p=\infty$ correspond to $L^1$ functions and $L^\infty$ functions , respectively.

Examples:

• On $((0,1),\mathcal{B},\lambda)$, the function $f(x)=x^{-1/2}$ lies in $L^1$ but not in $L^2$.
• On $(\mathbb{R},\mathcal{B},\lambda)$, the indicator function $\mathbf{1}_{[0,1]}$ lies in $L^p$ for every $1\le p\le\infty$, with $\|\mathbf{1}_{[0,1]}\|_p=1$ for $p<\infty$ and $\|\mathbf{1}_{[0,1]}\|_\infty=1$.