The Perron-Stieltjes integral

Ward, A. J.

doi:10.1007/bf01180442

Cited by 38 publications

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“…Our main result (Theorem 1.1) is proved using the Young-Stieltjes integral (see Subsection 2.2 for the definition). The integral representation also holds for two other extensions of the Riemann-Stieltjes integral: for the central Young integral, which was suggested by L. C. Young [39], and further modified by Dudley [5], as well as for the variant of the Perron-Stieltjes integral defined by Ward [38], and further developed by Kurzweil [17] and Henstock [12]. The p-variation of f is used to control its roughness.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Chain rules and p-variation

Norvaiša

2002

Studia Math.

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Abstract.The main result is a Young-Stieltjes integral representation of the composition φ • f of two functions f and φ such that for some α ∈ (0, 1], φ has a derivative satisfying a Lipschitz condition of order α, and f has bounded p-variation for some p < 1 + α. If given α ∈ (0, 1], the p-variation of f is bounded for some p < 2 + α, and φ has a second derivative satisfying a Lipschitz condition of order α, then a similar result holds with the Young-Stieltjes integral replaced by its extension.

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Section: Introduction and Resultsmentioning

confidence: 99%

Chain rules and p-variation

Norvaiša

2002

Studia Math.

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“…The integral of Perron [25] from 1914 uses families of major and minor functions instead of a single antiderivative. A "weighted" analogue of the Perron integral is the Perron-Stieltjes integral introduced by Ward [32]. In 1957, Kurzweil [18] introduced a gauge generalized Riemann type integral, which is equivalent to the Perron integral.…”

Section: Introductionmentioning

confidence: 99%

An elementary way to introduce a Perron-like integral

Bendová¹,

Malý²

2011

Ann. Acad. Sci. Fenn. Math.

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Abstract.We give an alternative definition of integral at the generality of the Perron integral and propose an exposition of the foundations of integral theory starting from this new definition. Both definition and proofs needed for the development are unexpectedly simple. We show how to adapt the definition to cover the multidimensional and Stieltjes case and prove that our integral is equivalent to the Henstock-Kurzweil(-Stieltjes) integral.

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“…AS in the proofs of A. J. Ward [49], page 581, Lemma 2, and page 592, Lemma 6, if g, y are VBG* there are g y , y y -(j -1,2) such that g, > 0, y, > 0, and g 2 , y 2 are finite and monotone increasing, and for small enough intervals, (46) lA.g^g.A.gj, |A 2 y|<y,A 2 y 2 .…”

Section: {S) F{p(t) -P(u) -F(u)(g(t) -G(u))} Dn + (U V T) (44) (S)mentioning

confidence: 99%

“…The problem ought to be easy even when the integrals are not absolute, but the changing of sign of / and <p causes difficulties. A quick proof notes several integral equivalences, namely, the Perron integral is equivalent to the Ward integral, to the variational integral, and to the Riemann-complete integral, a form of the generalized Riemann integral, see [49,21,25]. Then [28], page 83, Theorem 11, with the…”

Section: Of the Real Line Respectively Then F(t) That Is (1) / F(t)mentioning

confidence: 99%

A problem in two-dimensional integration

Henstock

1983

J Aust Math Soc A

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If two functions of a real variable are integrable over two intervals, say of t, T, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and T. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Cesaro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof. [a, b], [a, /?] of the real line, respectively, then f(t) that is, (1) / f(t) 0.[26], page 63, Theorem 7, is more general, but the assumption of bounded variation shows that we are again dealing with absolute integrals. The problem ought to be easy even when the integrals are not absolute, but the changing of sign of / and

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The Perron-Stieltjes integral

Cited by 38 publications

References 0 publications

Chain rules and p-variation

Chain rules and p-variation

An elementary way to introduce a Perron-like integral

A problem in two-dimensional integration

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