The Preservation of Convergence of Measurable Functions Under Composition

Abstract

Let / be a real-valued measurable function on a measure space (S, ©, p.) and let <p be a Borel measurable function of a real variable. Then it is well-known that the composition <p of, defined by <p of(s) = 4>(f(s)), sES, is also measurable. Conversely, if 4> is not Borel measurable, then there exists a measurable function/ on some measure space such that <f> of is not measurable. We summarize these two remarks in the statement that a function <j> preserves measurability under composition if and only if it is Borel measurable. The purpose of this note is to characterize functions 4> which preserve convergence (in various senses) of measurable functions. After these results were obtained it was pointed out that Theorems 5 and 6 follow from theorems of P. R. Halmos [2]. Theorem 7 was proved by

Representation of additive and biadditive functionals

“…Such functionals also occur in the functional analytic study of ordinary differential equations. These and other applications will be dealt with elsewhere Apart from these applications the representation theorems obtained here are of intrinsic interest and provide generalizations of results established in Halmos [5] and Bartle and Joichi [7] concerning certain nonlinear operators on function spaces.…”

Mapping properties that preserve convergence in measure on finite measure spaces

“…It follows directly from the results in [1] that the implication (4) fails for infinite measure spaces, even if Y = Z = R and the mappings f n and f are measurable. However, it is also easy enough to give a short counterexample.…”

“…Indeed, this is proved in [1,Theorem 2] in the case where Y = Z = R and the functions f n and f are measurable, but the argument given there carries over with only minor notational changes to the metric-space case for arbitrary functions f n and f with "convergence in measure" replaced by "convergence in outer measure." Remark 6.…”

“…Remark 6. References [1,8] contain proofs of a result which shows that the implication (4) holds under the assumptions that (X, M, μ) is a finite measure space, Y = Z = R, the mappings f n and f are measurable, and g : R → R is continuous (see [1,Theorem 1] and [8, p. 102]). As mentioned at the beginning of this note, this result is also stated as an exercise in [2,5].…”

Random walks and induced Dirichlet forms on self-similar sets

“…However this requires that χ Sx(K) (ι(Z τ ℓ )) → χ Sx(K) (ι(Z ∞ )) a.e., but we cannot make such conclusion directly, as the convergence for the composition f (ι(Z τ ℓ )) generally requires the continuity of f [4]. We need to go through a more detailed analysis.…”