Stieltjes Integral meaning.

You can imagine $\alpha(\xi)$ as a non-uniform scaling applied to the $x$-axis.

Imagine, for example, that you're driving through the andes in peru, and your car's fancy GPS records the current elevation every kilometer (or mile if you prefer to keep it imperial). A few days later you compete in a marathon, and are asked to disclose the average altitude you were on, since your blood test shows an elevated count of red blood cells which is an indication of doping unless you spent a considerable amount of time at high altitudes.

Easy, you think, and download the data from your GPS, compute the sum, and divide by the number of data points. You report that value, and a prompty disqualified from this and any further events. What happened? It turns out your average speed was much higher at low altitudes than it was at high ones - perfectly reasonable, given the condition of some of the streets in those mountainous regions. Thus, you actually spent much more times at higher altitudes than the graph of altitude over distance that your GPS record shows. What you need is a graph of altitude over time, but you don't have that.

When you discover the problem, you check your car's logs as well, and are delighted to discover that your car maintens a log of the elapsed time, also recorded every kilometer (or mile). Can I use this to compute a corrected average, you ask a friend of yours, and thus rehabilitate my good name in the runner community? You can, he answers, and for once you're glad that you have a mathematician amongst your friends...

What does your mathematically inclined friend do? He aligns the list of altitudes from your GPS with the list of times from your car, and gets a table like the following $$ \begin{array}{lll} \textrm{Distance} & \textrm{Altitude} & \textrm{Time} \\ \hline \\ 0\textrm{km} & 500\textrm{m} & 0\textrm{s} \\ 1\textrm{km} & 550\textrm{m} & 30\textrm{s} \\ 2\textrm{km} & 600\textrm{m} & 65\textrm{s} \\ \ldots \end{array} $$

He then computes a new column "Time Delta" which contains the difference between each time and the one that immediatly precedes it. The new table looks like this $$ \begin{array}{lll} \textrm{Distance} & \textrm{Altitude} & \textrm{Time} & \textrm{Time Delta} \\ \hline \\ 0\textrm{km} & 500\textrm{m} & 0\textrm{s} & - \\ 1\textrm{km} & 550\textrm{m} & 30\textrm{s} & 30\textrm{s} \\ 2\textrm{km} & 600\textrm{m} & 65\textrm{s} & 35\textrm{s} \\ \ldots \end{array} $$

Each time delta then reflects the time it took to drive 1 kilometer (or mile), i.e. it reflects the velocity you were driving with (altough in seconds/kilometer, not kilometers/hour as usual. You could call that "slowness" instead of "speed", because while it refelects the velocity, it reflects higher velocities with lower values, and lower velocities with higher values). Each Time Delta also reflects the time you spend at the given altitude, since it's simply the time between the last altitude measurement and the current one.

All your friends needs to do then is to again average the altitude values, but weighted with their corresponding time deltas. In other words, he simply multiplies each altitude with the corresponding time delta, adds up all those producs, and divides by the sum of the time deltas (which is simply the delta between the first and the last logged time). And voila, the resulting average is much higher than the one you got, and upon notifying the authorities your ban is lifted and your name is cleared.

Stiltjes Integrals do exactly this kind of weighted summation. In the Stiltjes integral $$ \int f(\xi) \,d\alpha(\xi) $$ $f(\xi)$ corresponds to the mapping from distance to altitude that your GPS gave you, while $\alpha(\xi)$ corresponds to the mapping of distance to time that your car logged. The integral weights each $f(\xi)$ with the rate of change of $\alpha$ at $\xi)$, i.e. it computes $$ \int f(\xi) \alpha'(\xi) d\xi \text{.} $$ Note how, compared to the tables above, the computation of differences between successive values has been replaced by taking the derivative. This is necessary since when $x$ ranges over a continuum, there's no longer such a thing as an immediate predecessor of a certain $x$.

Beware that this formula is only correct if $\alpha(\xi)$ is continously differentiable!

$$\int_a^b \xi^3\,d\alpha(\xi)=\int_a^b\xi^3a'(\xi)d\mu(\xi) $$ in case that $a$ is differentiable. Also $\mu(\xi) $ indicates the Legesgue measure. In order to find the above formula, you have to use Radon-Nikodym derivatives. (http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem).

Generally the function $a$ gives you a way to measure elementary sets, such as intervals. This function will produce the corresponding Stieltjes measure. In the same way, measuring intervals(or cubes) with the natural way ($\mu([0,1])=1)$, produces Lebesgue measure.