FubiniTonellitype

FubiniTonellitype is a term used in some measure-theory discussions to classify measurable functions on product spaces that satisfy the exchange of integration order in a unified way, drawing on the ideas behind Fubini's and Tonelli's theorems. It is not a universally adopted name in all texts, but appears in lightweight expositions and research notes as a convenient shorthand for a common set of integrability conditions.

Informally, a function f on a product measure space X × Y, with measures μ on X and ν on Y, is said to be of FubiniTonellitype if the standard criteria that guarantee interchange of integrals are satisfied in a single, coherent framework. A typical formulation mirrors the classical results: if f is nonnegative or if ∫∫|f| dμdν < ∞, then the iterated integrals ∫(∫ f(x,y) dν(y)) dμ(x) and ∫(∫ f(x,y) dμ(x)) dν(y) exist and equal the double integral ∫∫ f dμdν. In many presentations, this type also requires that the sections f_x(y) and f^y(x) be measurable and integrable as appropriate, and that the inner integrals define measurable functions on the remaining variable.

Relationship to standard theorems: the FubiniTonellitype umbrella encompasses both Tonelli’s theorem (for nonnegative functions) and Fubini’s theorem (for integrable functions) under a unified naming scheme. The concept aims to highlight a shared set of hypotheses that ensure the equality of iterated and double integrals across a broad class of functions.

See also: Fubini's theorem, Tonelli's theorem, iterated integrals, product measure, σ-finiteness.