L1 function

L^1 is the collection of all absolutely integrable functions, packaged as a single space you can measure distances in. The size of a function is its total absolute area, the integral of |f|, called its L^1 norm. Two functions count as the same point if they agree almost everywhere — we glue together functions differing only on a measure-zero set, because the integral cannot tell them apart.

Formally, fix a measure space; L^1(μ) is the set of (equivalence classes of) measurable functions f with the integral of |f| dμ finite, with norm equal to that integral. The almost-everywhere identification is essential: without it the norm would be only a seminorm, since a nonzero function vanishing a.e. would have zero norm. With the identification, the norm is genuine and the distance between f and g is the integral of |f - g|.

The deep fact is that L^1 is complete: every Cauchy sequence in this norm converges to an L^1 function. So L^1 is a Banach space — this is the Riesz–Fischer theorem for p = 1. Completeness is exactly what fails for the Riemann integral with this norm, and it is the main structural reason the Lebesgue theory replaced the Riemann theory in analysis.