1. ## Riemann Stieltjes

Let $\alpha$ be an increasing function on $[a,b], \quad f,g \in \mathcal{R}[\alpha;[a,b]]$ and g is non negative on [a,b]. Suppose $m= \inf \{f(x): a \leq x \leq b \}$ and $M= \sup \{f(x) : a \leq x \leq b\}$. Show that there is a $\lambda \in [m,M]$ such that $\int\limits_{a}^{b} fg \ d\alpha = \lambda \int\limits_{a}^{b} g \ d\alpha$

2. Originally Posted by Chandru1
Let $\alpha$ be an increasing function on $[a,b], \quad f,g \in \mathcal{R}[\alpha;[a,b]]$ and g is non negative on [a,b]. Suppose $m= \inf \{f(x): a \leq x \leq b \}$ and $M= \sup \{f(x) : a \leq x \leq b\}$. Show that there is a $\lambda \in [m,M]$ such that $\int\limits_{a}^{b} fg \ d\alpha = \lambda \int\limits_{a}^{b} g \ d\alpha$
What have you tried? Also, are there stipulations missing?

3. ## Attempt

Hi-

I thought of applying the Mean Value theorem. But i dont know,...i am not geting it. Please help me out.

4. Originally Posted by Chandru1
Hi-

I thought of applying the Mean Value theorem. But i dont know,...i am not geting it. Please help me out.
Well...if you know the mean-value theorem for integrals...this is kind of immediate. Do you?