Riemman-Stieltjes Integral: Proving Supremum Property

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In summary, the Riemann-Stieltjes integral is a powerful and flexible integration method that takes into account the behavior of both the function being integrated and the interval. The supremum property is a key aspect of this integral, ensuring its validity and allowing for the development of the fundamental theorem of calculus. This property is used to show that the upper and lower Riemann-Stieltjes sums converge, proving the existence and uniqueness of the integral. While the supremum property can be applied to any function and interval, it does have some limitations, such as requiring the function to be integrable and the interval to meet certain conditions.
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takeuchi
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Let α>0, J:=[-a,a] and f:J→ℝ a bounded function.
Let α an increases monotonically on J and P* the set of all partitions P of J containing 0 and such that are symmetric,i.e, x in P iff -x in P. Prove that
∫fdα= sup L(P,f,α) with P in P*
 
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It is sufficient to show that for a given e>0, there exist a P* such that ∫fdα-L(P*,f,α)<e.
Find a P first and let P* be a refinement of P, which is also symmetric and containing 0. Then P* is what you want.
 

Related to Riemman-Stieltjes Integral: Proving Supremum Property

1. What is the Riemann-Stieltjes integral?

The Riemann-Stieltjes integral is a generalization of the Riemann integral, which is used to calculate the area under a curve. It takes into account not only the function being integrated, but also another function that defines the behavior of the interval. This allows for a more flexible and powerful integration method.

2. What is the significance of the supremum property in the Riemann-Stieltjes integral?

The supremum property is a key property of the Riemann-Stieltjes integral that ensures its validity. It guarantees that the integral can be approximated by Riemann sums, which are easier to calculate. This property also allows for the development of the fundamental theorem of calculus, which relates differentiation and integration.

3. How is the supremum property used to prove the validity of the Riemann-Stieltjes integral?

The supremum property is used to show that the upper and lower Riemann-Stieltjes sums converge to the same value as the interval becomes finer. This is done by using the definition of the supremum and infimum of a set of numbers. By showing that these sums converge, it can be proven that the integral exists and has a unique value.

4. Can the supremum property be applied to any function and interval?

Yes, the supremum property is a general property that can be applied to any function and interval. It is not limited to specific types of functions or intervals, making it a powerful tool in the evaluation of integrals.

5. Are there any limitations to the use of the supremum property in the Riemann-Stieltjes integral?

One limitation of the supremum property is that it requires the function to be integrable. This means that it must be bounded and have a finite number of discontinuities. Additionally, the function defining the interval must also satisfy certain conditions. If these conditions are not met, the supremum property cannot be used to prove the validity of the integral.

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