 Lp space

For a measure space $(X,\mathcal{F},\mu )$ and $1\le p<\infty$, $L^p(\mu )$ consists of equivalence classes of measurable functions satisfyingTwo representatives are identified when they agree almost everywhere. For $p=\infty$, use the essential supremum norm, not an infinite power inside the integral. The quotient is essential: a function supported on a null set has zero integral norm even if it is not identically zero.^[ref]

These are Banach spaces; L2 space also has the integral inner product making it a Hilbert space. On $(0,1)$ with Lebesgue measure, $f(x)=x^{-\alpha }$, $\alpha >0$, belongs to $L^p$ exactly when $\alpha p<1$. This shows why membership depends on the exponent: a function can be integrable while its square is not. For $0<p<1$, the same power expression defines a quasi-norm rather than a norm, as explained under Lq quasi-norm.

In functional analysis, an \( L^p \) space (or Lebesgue \( p \)-space) is a vector space of measurable functions for which the \( p \)-th power of the absolute value is integrable.

Integrable functions to the power $p$, usually and in this text assumed under the Lebesgue integral because: Lebesgue integral of $L^p$ is complete but Riemann isn't