measurable,
adj. 1a. (of a set) belonging to the sigma-algebra in a given measurable space.
b. (of a set A with respect to a given outer measure μ*) having the property that for every subset E of the space
μ*(A) = μ*(A ⊕ E) + μ*(A \ E). 2. (of a function or transformation between measure algebras) such that the inverse image of a measurable set in the range space is measurable in the domain space. Thus a real-valued function is Borel measurable if the inverse image of every open (or Borel) set is Borel measurable, and a real-valued function is Lebesgue measurable if the inverse image of every open (or Borel) set is Lebesgue measurable. When the range of f is the extended real numbers, f-1(± ∞) is also required to be measurable. Intuitionists hold that all sets are Lebesgue measurable since the construction of non-measurable sets depends upon the axiom of choice, and it has been shown that the assumption that all sets are Lebesgue measurable is consistent with the remaining axioms of set theory.
b. (of a set A with respect to a given outer measure μ*) having the property that for every subset E of the space
μ*(A) = μ*(A ⊕ E) + μ*(A \ E). 2. (of a function or transformation between measure algebras) such that the inverse image of a measurable set in the range space is measurable in the domain space. Thus a real-valued function is Borel measurable if the inverse image of every open (or Borel) set is Borel measurable, and a real-valued function is Lebesgue measurable if the inverse image of every open (or Borel) set is Lebesgue measurable. When the range of f is the extended real numbers, f-1(± ∞) is also required to be measurable. Intuitionists hold that all sets are Lebesgue measurable since the construction of non-measurable sets depends upon the axiom of choice, and it has been shown that the assumption that all sets are Lebesgue measurable is consistent with the remaining axioms of set theory.