John J. Coffey and Robert Merkovsky Purdue University Calumet Hammond, Indiana 46323

A Note on the Upper and Lower Henstock Integrals

Abstract

It is shown that the upper and lower Henstock integrals coincide with upper and lower Perron integrals, when the former exist.

1 Introduction

In [2], the notions of upper and lower Henstock integrals are defined using a set-valued map of an interval variable, called a contraction. In this paper, we show that when defined, these agree with the usual upper and lower Perron integrals.

The usual definition of the upper Perron integral is given in terms of major functions, see [5]. Pfeffer has shown that it can also be defined in terms of Riemann-Stieltjes sums, see [3] and [4]. See also [1].

2 Notation and Terminology

Throughout the paper, f will refer to a measurable function f : [a, b] → R, and δ will refer to a gauge δ : [a, b] → (0, +∞).

A Perron partition is a set P := {(t_{i} , [x_{i−1}, x_{i}]) : i = 1, … n }, where a = x_{0} ≤ x_{1} ≤ ... ≤ x_{n} = b and x_{i−1} ≤ t_{i} ≤ x_{i} for i = 1, … n.

We let I[a, b] denote the set of closed subintervals of [a, b]. We include singletons in this set. Let 2^{[a,b]} denote the set of all subsets of [a, b]. A contraction is a map ν : I[a, b] → 2^{[a,b]} having the following three properties:

C1. For each J ∈ I[a, b], v(J) ⊆ J.

C2. For each J_{1}, J_{2}, ∈ I[a, b], J_{1} ⊆ J_{2} implies that ν(J_{2}) ∩ J_{1} ⊆ ν(J_{1}).

C3. ∪{ν(J)∩J^{∘}} = [a, b], where J^{∘} is the interior of J in the relative topology of [a, b].

For any gauge δ[a, b] → (0, ∞), define a mapping ν_{δ} : I[a, b] → 2^{[a,b]}, called the gauge contraction corresponding to δ by ν_{δ} : J ↦ {x ∈ J : J ⊂ (x − δ(x), x + δ(x))}. It is an easy exercise to verify that ν_{δ} is a contraction.

Let ν : I[a, b] → 2^{[a,b]} be a contraction and let P = {(t_{i} , [x_{i−1}, x_{i}]) : i = 1, … n } be a Perron partition. We say that P is ν − fine if ν([x_{i−1}, x_{i}]) ≠ ∅ for i = 1, …, n. We note in passing that the tags of P are irrelevant to this definition. However, given a gauge δ , it is easy to see that if P is δ- fine, then P is ν_{δ}- fine.

If ν_{1} and ν_{2} are contractions, we say that ν_{1} ≺ ν_{2} if for every closed interval I, ν_{1}(I) ⊇ ν_{2}(I). It is easy to see that if ν_{1} ≺ ν_{2}, and P is ν_{1}-fine, then P is ν_{2}-fine.

Let f : [a, b] → R be a function. Let ν : I[a, b] → 2^{[a,b]} be a contraction and P = {(t_{i} , [x_{i−1}, x_{i}]) : i = 1, … n } be a ν-fine Perron partition. As in [2], we define the following analogues of lower Riemann sums for f bounded on ν([x_{i−1}, x_{i}]), i = 1, …, n.

S^{l}(P, f) = ∑_{i=1}^{n} inf{f(t) : t ∈ ν([x_{i−1}, x_{i}])} (x_{i} − x_{i−1})

S^{l}(ν, f) = inf_{P} S^{l}(P, f),

where the infimum is taken over the set of all ν-fine partitions P = {(t_{i} , [x_{i−1}, x_{i}]) : i = 1, …, n } .

The analogs of upper Riemann sums, S^{u}(P, f) and S^{u}(ν, f) are defined similarly using supremums in place of infimums. As is noted in [2] , S^{l}(ν, f) and S^{u}(ν, f) can be written in terms of a single infimum or supremum. In fact,

S^{l}(ν, f) = inf_{P} σ(P, f),

where the infimum is taken over the set of all ν-fine Perron partitions P = {(t_{i} , [x_{i−1}, x_{i}]) : i = 1, … n } . S^{u}(ν, f) is given by the same formula with inf replaced by sup. We remark that S^{l}(P, f), S^{l}(ν, f), S^{u}(P, f), and S^{u}(ν, f) are defined in [2] only when they are finite. It is shown in [2] that S^{l}(ν, f) is monotone increasing and S^{u}(ν, f) is monotone decreasing in ν.

The upper Henstock integral of f is defined to be:

(\overline{H}) ∫_{a}^{b} f(x)dx := inf_{ν} S^{u}(ν, f)

where the infimum is taken over the set of all contractions ν for which S^{u}(ν, f) is defined and ν_{0} ≺ ν, where ν_{0} is a fixed contraction for which S^{u}(ν_{0}, f) is defined. We follow [2] in declining to define this upper integral if the appropriate set of contractions is empty. The lower Henstock integral is defined analogously

(\underline{H}) ∫_{a}^{b} f(x)dx := sup_{ν} S^{l}(ν, f).

Major and minor functions are defined as in [5]. The upper Perron integral of f is defined to be

(\overline{P}) ∫_{a}^{b} f(x)dx := inf\{U_{δ}(b) : U is a major function for f(x)on [a, b]\}

The lower Perron integral of f is defined to be

(\underline{P}) ∫_{a}^{b} f(x)dx := sup\{U_{δ}(b) : U is a major function for f(x)on [a, b]\}

We adopt the conventions that sup ∅ = −∞ and inf ∅ = +∞. The Perron integral exists if and only if the upper and lower integrals are finite and equal.

Let f : [a, b] → R be a measurable function, and P a Perron partition. We use the following notation for the Riemann sum of f with respect to P

σ(P, f) := ∑_{i=1}^{n} f(t_{i})(x_{i} − x_{i−1}). Let δ : [a, b] → (0, +∞) be a gauge and P a

Perron partition. P is δ − fine if [x_{i−1}, x_{i}] ⊆ (t_{i} − δ(t_{i}), t_{i} + δ(t_{i})), i = 1, ..., n.

Let f : [a, b] → R be a measurable function, and δ : [a, b] → (0, +∞) be a gauge. We define the upper interval function U_{δ} : I[a, b] → R_{e} as follows. For each c ∈ [a, b], we define U_{δ}[c, c] = 0 and for a ≤ c < d ≤ b, we define

U_{δ}[c, d] = sup{σ(P, f) : P is a δ-fine partition of [c, d]}.

We note that U_{δ}[c, d] > −∞. We allow the possibility that U_{δ}[c, d] = +∞.

L_{δ}[c, d] is defined analogously, with the supremum replaced by infimum. We note that L_{δ}[c, d] < +∞. We allow the possibility that L_{δ}[c, d] = −∞.

Let δ_{1} and δ_{2} be gauges. We say that δ_{2} is a refinement of δ_{1} and write δ_{1} ≺ δ_{2} if for each x ∈ [a, b] we have δ_{1}(x) ≥ δ_{2}(x).

The following result was proven in [3].

Theorem 2.1 The upper and lower Perron integrals may be defined as:

2.1.1. (\overline{P}) ∫_{a}^{b} f(x)dx := inf\{U_{δ}[a, b] : δ is a gauge\}.

2.1.2. (\underline{P}) ∫_{a}^{b} f(x)dx := sup\{L_{δ}[a, b] : δ is a gauge\}.

3 Relation of Upper and Lower Perron Integrals to Upper and Lower Henstock Integrals

Lemma 3.1 For each contraction ν, there is a gauge δ such that ν ≺ ν_{δ}.

Proof. For each x ∈ [a, b], define

δ(x) := min{2, inf{max{x − c, d − x} : c ≤ x ≤ d and x ∉ ν([c, d])}}.

For clarity, we point out that x is being considered as fixed, and that the infimum is taken over all intervals [c, d] for which x ∈ [c, d]\ν([c, d]).

Clearly, 0 ≤ δ(x) ≤ 2 < +∞. In order to establish that δ is a gauge, we need only show that δ(x) > 0. Consider x to be fixed. By the third of the defining properties of a contraction, [C3], there is a subinterval, J, of [a, b] such that x ∈ ν(J) ∩ J^{∘}. Call J = [g, h].

We proceed by examining three cases. First, suppose that g ≠ a and h ≠ b. Then g < x < h. Assume without loss of generality that [c, d] is an interval for which x ∈ [c, d]\ν([c, d]). By way of contradiction, suppose [c, d] ⊆ [g, h], then by defining property [C2] of a contraction, ν[g, h] ∩ [c, d] ⊆ ν([c, d]). But x ∉ ν([c, d]) and x ∈ ν([g, h]) ∩ [c, d]. This contradiction shows that [c, d] is not a subset of [g, h]. Therefore, either h < d or g > c. Hence, either d − x > h − x or x − c > x − g. Therefore, δ(x) ≥ max(h − x, x − g) > 0.

The cases where g = a or h = b are handled similarly. Thus, δ is a gauge. In order to verify ν ≺ ν_{δ}, it suffices to show that ν_{δ}([g, h)] ⊆ ν([g, h]) for

each subinterval [g, h] of [a, b]. Let x ∈ ν_{δ}([g, h]). By definition of ν_{δ}, x ∈ [g, h] and [g, h] ⊆ (x − δ(x), x + δ(x)), or equivalently, δ(x) > max(h − x, x − g). Therefore, x ∈ ν([g, h]), and hence ν_{δ}([g, h]) ⊆ ν([g, h]).

Theorem 3.2 The upper Perron integral of a function is finite if and only if the upper Henstock integral exists. It this case, the two upper integrals have the same value. A similar assertion holds for the lower Perron and Henstock integrals.

Proof. We give the proof only for the upper integrals. The result for the lower integrals follows upon replacing the integrand by its negative. Assume

that (\overline{P}) ∫_{a}^{b} f(x)dx < +∞. Pick ϵ > 0. By Theorem 2.1, there is a gauge δ such

that

(\overline{P}) ∫_{a}^{b} f(x)dx ≤ U_{δ}[a, b] < (\overline{P}) ∫_{a}^{b} f(x)dx + ϵ.

It follows readily from the definitions that U_{δ}[a, b] = S^{u}(ν_{δ}, f). We may take the contraction, ν_{0}, in the definition of the Henstock integral to be ν_{δ}. Thus, the upper Henstock integral is defined, and moreover

(\overline{H}) ∫_{a}^{b} f(x)dx ≤ U_{δ}[a, b] < (\overline{P}) ∫_{a}^{b} f(x)dx + ϵ.

Letting ϵ go to zero, we obtain

(\overline{H}) ∫_{a}^{b} f(x)dx ≤ (\overline{P}) ∫_{a}^{b} f(x)dx.

Now assume that the upper Henstock integral of f is defined. Pick ϵ > 0. There is a contraction ν such that

(\overline{H}) ∫_{a}^{b} f(x)dx ≤ S^{u}(ν, f) < (\overline{H}) ∫_{a}^{b} f(x)dx + ϵ.

By Lemma 3.1, there is gauge δ such that ν ≺ ν_{δ}. Hence,

(\overline{P}) ∫_{a}^{b} f(x)dx ≤ S^{u}(ν_{δ}, f) ≤ S^{u}(ν, f) < (\overline{H}) ∫_{a}^{b} f(x)dx + ϵ.

Since ϵ is arbitrary, the result follows.

References

[1] P. S. Bullen and R. Vyborny, Some applications of a theorem of Marcinkiewicz, Canad. Math. Bull. 34 (1991), no. 2, 165 - 174.

[2] P.Y. Lee and Z. Dongsheng, Upper and lower Henstock integrals, Real Analysis Exchange 22 (1996/7), no. 2, 734-739.

[3] W.F. Pfeffer, The Riemann-Stieltjes approach to integration, TWISK 187:1980, National Research Institute for Mathematical Sciences, Pretoria (South Africa).

[4] W.F. Pfeffer, The Riemann Approach to Integration:Local Geometric Theory, Cambridge University Press, Cambridge, 1993.

[5] S. Saks, The Theory of the Integral (Second Edition, revised) Hafner Publishing Company, New York, 1937.