Let’s say we take an integrator 
of bounded variation on an interval ![\left[a,b\right]](http://s0.wp.com/latex.php?latex=%5Cleft%5Ba%2Cb%5Cright%5D&bg=e6e6e6&fg=333333&s=0 "\left[a,b\right]")
and a function 
that’s Riemann-Stieltjes integrable with respect to over that interval. Then we know that is also integrable with respect to over the subinterval ![\left[a,x\right]\subseteq\left[a,b\right]](http://s0.wp.com/latex.php?latex=%5Cleft%5Ba%2Cx%5Cright%5D%5Csubseteq%5Cleft%5Ba%2Cb%5Cright%5D&bg=e6e6e6&fg=333333&s=0 "\left[a,x\right]\subseteq\left[a,b\right]")
. Let’s use this to define a function 
on by
![\displaystyle F(x)=\int\limits_{\left[a,x\right]}fd\alpha](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%28x%29%3D%5Cint%5Climits_%7B%5Cleft%5Ba%2Cx%5Cright%5D%7Dfd%5Calpha&bg=e6e6e6&fg=333333&s=0 "\displaystyle F(x)=\int\limits_{\left[a,x\right]}fd\alpha")
We can immediately say some interesting things about this function. First of all, is, like , of bounded variation. Next, wherever is continuous, so is . Finally, if is increasing, then is differentiable wherever is differentiable and is continuous. At such points, we have 
. Notice that, as usual, the first two results will hold if we show that they hold for increasing integrators.
These results are similar to those we get from the Fundamental Theorem of Calculus, and we can use some of the same techniques to prove them. In particular, we call on the Integral Mean-Value Theorem for Riemann-Stieltjes integrals. If we take points 
and 
in , this tells us that
![\displaystyle F(y)-F(x)=\int\limits_{\left[a,y\right]}fd\alpha-\int\limits_{\left[a,x\right]}fd\alpha=\int\limits_{\left[x,y\right]}fd\alpha=f(x_0)\left(\alpha(y)-\alpha(x)\right)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+F%28y%29-F%28x%29%3D%5Cint%5Climits_%7B%5Cleft%5Ba%2Cy%5Cright%5D%7Dfd%5Calpha-%5Cint%5Climits_%7B%5Cleft%5Ba%2Cx%5Cright%5D%7Dfd%5Calpha%3D%5Cint%5Climits_%7B%5Cleft%5Bx%2Cy%5Cright%5D%7Dfd%5Calpha%3Df%28x_0%29%5Cleft%28%5Calpha%28y%29-%5Calpha%28x%29%5Cright%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle F(y)-F(x)=\int\limits_{\left[a,y\right]}fd\alpha-\int\limits_{\left[a,x\right]}fd\alpha=\int\limits_{\left[x,y\right]}fd\alpha=f(x_0)\left(\alpha(y)-\alpha(x)\right)")
where 
is some point between 
and .
Now we let 
be the supremum of 
on . For any partition of we calculate the variation
thus giving an upper bound to the variation of .
Similarly, let be a point where is continuous. Then given an 
we can find a 
so that 
implies that 
. Thus 
, and we see that is continuous at as well.
Finally, we can divide by 
to find
Then as gets closer to , gets squeezed in towards as well. If is continuous at and is differentiable there, then the limit of this difference quotient exists, and has the value stated.
March 31, 2008
Posted by John Armstrong |
Analysis, Calculus |
2 Comments
