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 [ α , β ] and g ( [ α , β ] ) ⊂ [ a , b ] , then f ○ g is Riemann integrable (resp. measurable) on [ α , β ] . A well-known fact, on the other hand, states that f ○ 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 ∈ V ∖ { 0 } and g ∈ W ∖ { 0 } , f ○ 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 ∈ W ∖ { 0 } there exists a c-dimensional space V of measurable functions such that f ○ g is not measurable for all f ∈ V ∖ { 0 } .