Riemann Steiljes Integral when alpha changes

Question

For example if the question was following

$$ \int_{0}^2 x\,d \alpha $$

where $ \alpha (x) = x $ if $ 0\le x\le 1 $ and $ \alpha(x) = 3x $ when $ 1< x \le 2 $

Is it correct to solve it using

$$ \int_{0}^2 x d \alpha = \int_{0}^1 x d x + \int_{1}^2 x d (3x)? $$

If that is possible, does it mean that the value of $\alpha$ at $1$ is does not change the answer say if it was

$ \alpha (x) = x $ if $ 0\le x< 1 $ and $ \alpha(x)=3x $ when $ 1\le x\le 2 $ ?

Thank you! so as I said even if the value of $\alpha$ at the point of discontinuity was to change it wouldn't have an impact right? — Infinity, Dec 15 '13 at 08:16
Yes, you are right. Have you studied any theorem related to this? — Mhenni Benghorbal, Dec 15 '13 at 08:18
I have studied only the Reimann integrable criteria and if it is integrable how to find it using the Riemann sum as $ lim_{||P||\to 0} \sum_{i=0}^n f(t_i) \Delta aplha_i $ and I wasn't sure if an end pint changed whether the integral changes — Infinity, Dec 15 '13 at 08:29
Note that, you need to add $\alpha(1+)-\alpha(1-)$ to your answer. Or, alternatively, just use the theorem I gave you. See the posted answer. — Mhenni Benghorbal, Dec 15 '13 at 08:53

score 5 · Accepted Answer · answered Dec 15 '13 at 08:36

No, you must take account of the jump discontinuity of $\alpha$ at $x = 1$. The correct evaluation should be

\begin{align*} \int_{0}^{2} x \, d\alpha(x) &= \int_{0}^{1} x \, dx + 1 \cdot \{ \alpha(1+) - \alpha(1-) \} + \int_{1}^{2} x \, d(3x) \\ &= \left[ \frac{x^{2}}{2} \right]_{0}^{1} + 1 \cdot (3 - 1) + \left[ \frac{3x^{2}}{2} \right]_{1}^{2} \\ &= \frac{1}{2} + 2 + \frac{9}{2} = 7. \end{align*}

Or as Mhenni Benghorbal pointed out, you can apply the Riemann-Stieltjes version of the integration by parts as follows:

\begin{align*} \int_{0}^{2} x \, d\alpha(x) &= \left[ x \alpha(x) \right]_{0}^{2} - \int_{0}^{2} \alpha(x) \, dx \\ &= 2\alpha(2) - \int_{0}^{1} x \, dx - \int_{1}^{2} 3x \, dx \\ &= 12 - \left[ \frac{x^{2}}{2} \right]_{0}^{1} -\left[ \frac{3x^{2}}{2} \right]_{1}^{2} \\ &= 12 - \frac{1}{2} - \frac{9}{2} = 7. \end{align*}

Riemann Steiljes Integral when alpha changes

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