yhteismitallisia

Yhteismitallisia is a Finnish mathematical term used to describe a collection of functions or random variables that are jointly measurable. In measure theory and probability, the concept concerns how a map from a sample space to a product space behaves with respect to the relevant sigma-algebras. The term is formed from yhteis- meaning joint or common, mitallinen meaning measurable.

Formally, let Ω be a sample space with sigma-algebra F, and let X1, X2, ..., Xn be random variables from Ω to the real numbers. The collection {X1, X2, ..., Xn} is jointly measurable if the map X: Ω → R^n given by X(ω) = (X1(ω), ..., Xn(ω)) is measurable with respect to F and the Borel sigma-algebra on R^n. Equivalently, each coordinate Xi is a measurable function, and, for finite families, joint measurability is achieved when all Xi are measurable. For infinite families, joint measurability is a stronger condition than individual measurability.

Joint measurability is essential for defining joint distributions, expectations of functions of several variables, and integrals over multiple random variables. It also underpins the rigorous formulation of stochastic processes and their sample-path properties; for example, a process {X_t} is often required to be jointly measurable in time and outcome to ensure the well-definedness of quantities like E[f(X_t, t)]] or to study continuity and integrability.

Etymology and related concepts: the Finnish term reflects the idea of measuring several quantities together. Related concepts include product sigma-algebras, measurable mappings, joint distributions, and independence, each playing a role in how joint measurability is used in theory and applications.