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[tex]

\int_a^b f(x) d\alpha(x)

[/tex]

when the function [itex]\alpha(x)[/itex] is

**not**monotonically increasing on [itex][a,b][/itex]? Rudin only seems to define R-S integrals with respect to monotonically increasing functions, but there are sources I've found on the Internet that seem to imply this requirement is optional (some of them have made noises about [itex]\alpha[/itex] only needing to be a function of bounded variation)...what are the bare minimum requirements on [itex]\alpha[/itex] for the above integral to make sense?