What Are Riemann-Stieltjes Integrals and How Can We Visualize Them?

[Artusartos]

I'm having trouble visualizing the riemann-stieltjies integral...

Our textbook states:

We assume throughout this section that F is an increasing function on a closed interval [a,b]. To avoid trivialities we assume F(a)<F(b). All left-hand and right-hand limits exist...We use the notation

$$F(t^-)= lim_{x \rightarrow t^-} F(x)$$ and $$F(t^+)= lim_{x \rightarrow t^+} F(x)$$

For a bounded function f on [a,b] and a partition $$P={a=t_0 < t_1 < ... < t_n = b}$$ of [a,b], we write

$$J_F(f,P) = \sum_{k=0}^n f(t_k) [F(t_k^+) - F(t_k^-)]$$

The upper Darboux-Stieltjes sum is

$$U_F(f,P) = J_F(f,P) + \sum_{k=1}^n max(f, (t_{k-1}, t_k) [F(t_k^+) - F(t_{k-1}^-)]$$

I'm having trouble visualizing this...also, by F(x), do they mean the integral of f(x)?

Thanks in advance

 

[lurflurf]

No F (a confusing choice of variable) is a function that determines the size of intervals. In a Riemann integral
∫f dx
the interval [a,b] has size b-a
In a Riemann-Stieltjies integral
∫f dF
the interval [a,b] has size F(b⁺)- F(a^-)

and of course when F(x)=x the Riemann-Stieltjies integral reduces to the Riemann integral

This is helpful in many ways.
-We can take sums as a type of integral and unify sums and integrals
-We can have impulse function like the Dirac delta function which concentrate a change to a single point.

 

[Artusartos]

No F (a confusing choice of variable) is a function that determines the size of intervals. In a Riemann integral
∫f dx
the interval [a,b] has size b-a
In a Riemann-Stieltjies integral
∫f dF
the interval [a,b] has size F(b⁺)- F(a^-)

and of course when F(x)=x the Riemann-Stieltjies integral reduces to the Riemann integral

This is helpful in many ways.
-We can take sums as a type of integral and unify sums and integrals
-We can have impulse function like the Dirac delta function which concentrate a change to a single point.

Thanks a lot...is it ok if you give an example or something? So I can understand the difference better (between usual Reimann integrals and Reimann-Stieltjes Integrals)?

 

[Stephen Tashi]

is it ok if you give an example or something?

Something short of a complete example:

I see from your other posts that you know something about statistics. Suppose we have a random variable X whose distribution is defined by the statement:

There is a 0.3 probability that X = 0.5 and if X is not equal to 0.5 then the other possibilities for X are uniformly distributed on the intervals [0,0.5) and (0.5, 1].

How would you compute the expected value a function f(X) ? ( e.g. the case f(X) = X would be the expected value of X). I think the common sense way is;

$\bar{f(x)} = (0.3) f(0.5) + (1.0 - 0.3) \ ( \ (0.5) \int_0^{0.5} f(x) u_1(x) dx + (0.5) \int_{0.5}^{1} f(x) u_2(x) dx\ )$

Where $u_1(x)$ is the uniform distribution on [0,0.5) and $u_2(x)$ is the uniform distribution on (0.5, 1] and the integrals are Riemann integrals.

It would be convenient to define a single distribution function for X and write $\bar{f(x)}$ as a single integral (even if the practical computation of that integral amounted to the work above). However, a Riemann integral can't handle the "point mass" probability at X = 0.5 because, in a manner of speaking, it sits on a rectangle whose base has zero length.

From the viewpoint of probability theory, a Riemann-Stieljes integral can be regarded as way of defining a new form of integration that handles such "point masses". ( You can define a nondecresasing function $F(x)$ which has a jump of size 0.3 at x = 0.5 )

 

[blue_raver22]

for any help!

I understand that mathematical concepts can often be difficult to visualize, especially when dealing with abstract concepts like integrals. Let me try to explain the Riemann-Stieltjes integral in a more visual way.

First, let's define the notation used. F is an increasing function, meaning that as x increases, F(x) also increases. For our purposes, we can think of F as representing the "accumulation" of some quantity over the interval [a,b]. For example, if F(x) represents the cumulative rainfall over a period of time, then F(t) would be the total rainfall up until time t.

Now, imagine that we have a bounded function f(x) on the interval [a,b]. This function could represent anything, such as the velocity of a moving object over time. We want to find the Riemann-Stieltjes integral of f with respect to F, which we write as ∫f dF.

To calculate this integral, we first divide the interval [a,b] into smaller subintervals, represented by the partition P. Each subinterval has a left and right endpoint, denoted by t_k^- and t_k^+ respectively. We then take the value of f at each of these endpoints and multiply it by the difference in the cumulative values of F at those points (F(t_k^+) - F(t_k^-)). This is what is represented by J_F(f,P) in the notation above.

To get a better understanding, let's consider our example of rainfall. If we divide the time interval into smaller subintervals, we can imagine that at each time point t_k, we are taking the amount of rainfall during that time (f(t_k)) and multiplying it by the change in total rainfall from the beginning of the interval until that point (F(t_k^+) - F(t_k^-)). This gives us a "rectangular" area, which represents the contribution of that subinterval to the overall integral.

However, this method only takes into account the cumulative values of F at the endpoints of each subinterval. To get a more accurate approximation, we also consider the maximum value of f within each subinterval and multiply it by the difference in cumulative values of F at the endpoints of that subinterval. This is represented by the term max(f, (t_{k-1}, t_k)) in the notation above.

Overall, the Riemann-St