Question

Assume that 𝛼 is increasing on [𝑎, 𝑏]. Note that if 𝑓 ∈ 𝑅(𝛼) on [𝑎, 𝑏], then 𝑓

Answer 1

… then, $\displaystyle |f|\in R(\alpha) \text{ on }[a,b]\text{ and we have the inequality;}$

$\displaystyle \left|\int_a^bf(x)\ d\alpha(x)\right|\leq\int_a^b|f(x)|\ d\alpha(x).$

$\displaystyle \color{red}\text{This is a standard theorem.}$