nullequivalence

Nullequivalence is the equivalence relation on the set of measurable functions defined with respect to a measure space (X, Σ, μ) by f ~ g if the set {x ∈ X : f(x) ≠ g(x)} has μ-measure zero. It is commonly called almost everywhere equality, since two functions related in this way agree at all points outside a μ-null set. The relation is reflexive, symmetric, and transitive, and thus partitions the space of measurable functions into equivalence classes of functions that agree almost everywhere.

The quotient space formed by this relation is denoted by M/~, i.e., the set of equivalence classes [f]. In probability theory, random variables are typically treated as equivalence classes under almost sure equality. In functional analysis, nullequivalence underlies the construction of Lp spaces: the Lp space Lp(X, Σ, μ) consists of equivalence classes [f] of measurable functions with finite p-th power integral, with the norm defined by ||[f]||p = (∫ |f|^p dμ)^{1/p}. This norm is well-defined because replacing f by an a.e. equal function does not change the integral.

Examples illustrate the idea: a function that is zero everywhere except on a μ-null set is equivalent to the zero function, and indicators χA and χB are equivalent if μ(A Δ B) = 0 (the symmetric difference has measure zero). Variants exist for extended real-valued functions and on spaces where the measure or σ-algebra changes. Nullequivalence thus provides a robust notion of equality that ignores behavior on sets deemed negligible by the measure, enabling meaningful quotient constructions and analysis. See also almost everywhere, measure theory, Lp spaces, and equivalence relations.