Question

Let 𝛼 be increasing and 𝑓 ∈ 𝑅(𝛼) on [𝑎, 𝑏]. What condition can we impose on 𝑓

so that the given equality holds?

𝑏𝑏

|∫ 𝑓(𝑥) 𝑑𝛼(𝑥)| = ∫ |𝑓(𝑥)| 𝑑𝛼(𝑥) 𝑎𝑎