prove that if f is continuous on [a,b] with $\displaystyle f(x) \geq 0 \ , \forall x \in [a,b] $ prove that if g is strictly increasing on [a,b]

with $\displaystyle \int_a^b f . dg =0 $ then f(x)=0 $\displaystyle \forall x \in [a,b] $

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- Feb 28th 2010, 01:11 AMflower3prove...
prove that if f is continuous on [a,b] with $\displaystyle f(x) \geq 0 \ , \forall x \in [a,b] $ prove that if g is strictly increasing on [a,b]

with $\displaystyle \int_a^b f . dg =0 $ then f(x)=0 $\displaystyle \forall x \in [a,b] $ - Feb 28th 2010, 03:25 AMLaurent
The proof is the same as for usual Riemann integral. Procede by contradiction: assume that $\displaystyle f$ is not identically zero on $\displaystyle [a,b]$. Using continuity, prove that there exists $\displaystyle \epsilon>0$ and $\displaystyle a\leq u<v\leq b$ such that $\displaystyle f(x)\geq \epsilon$ when $\displaystyle x\in[u,v]$. Then, justify the following: $\displaystyle \int_a^b f\, dg\geq \int_u^v f\, dg\geq \epsilon \int_u^v dg=\epsilon(g(v)-g(u))>0$. And conclude.