Riemann integrability and Lebesgue measurability of the composite function

Abstract

If f is continuous on the interval [a, b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [alpha, beta] and g([alpha, beta]) subset of [a, b], then f o g is Riemann integrable (resp. measurable) on [alpha, beta]. A well-known fact, on the other hand, states that f o g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a 2(c)-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f is an element of V \ {0} and g is an element of W \ {0} . f o g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g is an element of W \ {0} there exists a c-dimensional space V of measurable functions such that f o g is not measurable for all f is an element of V \ {0}.