Let a C([a, b]) n BV([a, b]). Assume g = R(a) on [a, b] and define...

 

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Let a C([a, b]) n BV([a, b]). Assume g = R(a) on [a, b] and define B(x) = g(t) da(t) if x = [a, b]. Show that (a) if f is increasing on [a, b], there exists ro [a, b] such that [se JdB = f(a) " gia +f(B) [9 g da, [Hint: Write down the integration by parts formula for f f d and then apply the first mean value theorem to the integral df.] (b) if, in addition, f is continuous on [a, b], we also have I f(x)g(x) da(x) = f(a) "" gda + f(b) b) [9 g da. Let a C([a, b]) n BV([a, b]). Assume g = R(a) on [a, b] and define B(x) = g(t) da(t) if x = [a, b]. Show that (a) if f is increasing on [a, b], there exists ro [a, b] such that [se JdB = f(a) " gia +f(B) [9 g da, [Hint: Write down the integration by parts formula for f f d and then apply the first mean value theorem to the integral df.] (b) if, in addition, f is continuous on [a, b], we also have I f(x)g(x) da(x) = f(a) "" gda + f(b) b) [9 g da.