Assume [tex]$\alpha$[/tex] increases monotonically and [tex]$\alpha' \in \mathbb{R}$[/tex] on [tex]$[a, b]$[/tex]. (Here [tex]$\alpha'$[/tex] is the derivative of [tex]$\alpha$[/tex] with respect to [tex]$x$[/tex] in [tex]$[a, b]$[/tex]). Let [tex]$f$[/tex] be a bounded real function that maps [tex]$[a, b]$[/tex] into [tex]$\mathbb{R}^k$[/tex]. Then [tex]$f \in R(\alpha)$[/tex] if and only if [tex]$f\alpha' \in R$[/tex]. In that case, [tex]$\int_a^b f \, d\alpha = \int_a^b f(x) \alpha'(x) \, dx$[/tex].
Answer (2)
This question addresses a topic in mathematical analysis, specifically dealing with the integration of functions with respect to functions that are not necessarily strictly linear, often referred to as the Riemann-Stieltjes integral. ;
The question focuses on the equivalence between Riemann-Stieltjes and Riemann integrals, demonstrating that a bounded function's integrability with respect to a monotonically increasing function depends on the integrability of their derivative product. It allows calculating the integral using conventional methods when conditions are met. This concept is essential in advanced mathematics for simplifying complex integrals.
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