A NONABSOLUTELY CONVERGENT INTEGRAL DEFINED BY PARTITIONS OF THE UNITY *

Giuseppa Riccobono - Dipartimento di Matematica - Universitá di Palermo - Via Archirafi, 34 - 90123 Palermo -Italy–E-mail:ricco@dipmat.math.unipa.it

Abstract.

In a compact metric measure space with measure μ, we define an integral by partitions of the unity such that a μ−integrable function is also integrable and a function which is integrable but it is not μ−integrable is constructed in a suitable non Euclidean space.

Keywords: Lebesgue measure, partition of unity,(PU)^{∗}-integral

Classification: 28A25

Introduction

In [5], a type of integral defined by partitions of the unity (PU-integral) is defined on an abstract compact metric measure space and it is proved that a PUintegrable function is μ−integrable and that the μ−integral is equivalent to the PU-integral. Moreover an example of a non Euclidean space on which is defined this type of integral is given. The PU-integral is obtained by approssimations of type Riemann sums. The advantage to use a such integral is that it does not use the geometry of the space so it can be defined in any abstract space.

In this paper X denotes a compact metric space, M a σ-algebra of subsets of X such that each open set is in M, μ a non-atomic, finite, complete Radon measure on M such that:

α) each ball U(x, r) centered at x with radius r has a positive measure, β) for every x in X there is a number h(x) ∈ ℜ such that μ(U[x, 2r]) ≤

h(x) · μ(U[x, r]) for all r > 0 (where U[x, r]) is the closed ball), γ) μ(∂U(x, r)) = 0 where ∂U(x, r) is the boundary of U(x, r).

Definition 1 A partition of unity (PU-partition) in X is, by definition, a finite collection P = {(θ_{i}, x_{i})}_{i=1}^{p} where x_{i} ∈ X and θ_{i} are non negative, μmeasurable and μ-integrable real functions on X such that ∑_{i=1}^{p} θ_{i}(x) = 1 a.e. in X.

The PU-partition is a PU^{∗}-partition if x_{i} ∈ S_{θ_{i}} = {x ∈ X : θ_{i}(x) ≠ 0}.

Definition 2 If δ is a positive function on X, a PU-partition P = {(θ_{i}, x_{i})}_{i=1}^{p} is said to be δ-fine if S_{θ_{i}} ⊂ U(x_{i}, δ(x_{i})) (i = 1, 2, .., p).

Definition 3 A real function f on a compact set A ⊆ X is said to be (PU)- integrable on A if there exists a real number I with the property that, for every

* This work was supported by M.U.R.S.T. given ϵ > 0, there is a positive function δ such that | ∑_{i=1}^{p} f(x_{i}) · ∫_{A} θ_{i}dμ − I |< ϵ for each δ-fine (PU)-partition P = {(θ_{i}, x_{i})}_{i=1}^{p} in A.

The number I is called the (PU)-integral of f on A and we write I = (P U) ∫_{A} f.

For (PU)^{∗}-partitions, we have the (PU)^{∗}-integral and set I = (P U)^{∗} ∫_{A} f.

Main results

Proposition 1 If δ is a positive function on a compact set A ⊆ X then there is a δ-fine PU^{∗}-partition in A.

Proof . See the Proof of Proposition 1.1 in [5]. Denoting by PU^{∗}(A) the family of all the PU^{∗}-integrable real functions on A,

the following Proposition is an immediate consequence of the Definition 4.

Proposition 2 If A ⊆ X is compact, then:

1) PU^{∗}(A) is a linear space and the map f → (P U)^{∗} ∫_{A} f is a non negative linear functional on PU^{∗}(A);

2) if k ∈ ℜ and f(x) = k for each x ∈ A then f ∈ PU^{∗}(A) and (P U)^{∗} ∫_{A} f = kμ(A).

3) if f , g ∈ PU^{∗}(A) and f ≤ g then (P U)^{∗} ∫_{A} f ≤ (P U)^{∗} ∫_{A} g. If P = {(θ_{i}, x_{i})}_{i=1}^{n} is a partition in A, set σ(f, P) = ∑_{i=1}^{n} f(x_{i}) ∫_{A} θ_{i}dμ.

Proposition 3 If A is a compact subset of X and if f ∈ PU^{∗}(X) then f ∈ PU^{∗}(A).

Proof . See Proposition 1.3 in [5].

Proposition 4 If f is a real function on a compact set A ⊆ X, then f ∈ PU^{∗}(A) if and only if for each ϵ > 0 there is a positive function δ on A such that |σ(f, P) − σ(f, Q)| < ϵ for every P = {(θ_{i}, x_{i})}_{i=1}^{n} and Q = {(θ′_{i}, x′_{i})}_{i=1}^{p} δ−fine PU^{∗}-partitions in A.

Proof . See proposition 1.2 in [5].

Proposition 5 If f is μ−measurable and μ−integrable on X, then f ∈ PU^{∗}(X) and (P U)^{∗} ∫_{X} f = ∫_{X} fdμ.

Proof. It follows by the equivalence between the PU-integral and the μ−integral (see [5]) and because a PU^{∗}-partition is also a PU-partition.

Proposition 6 A PU^{∗}-integrable function is μ−measurable. Proof It is analogue to that used in [5] Propositions 2.3 and 2.4.

Proposition 7 If f, g are two real functions on X and f = g a.e. in X then g is (PU)^{∗}-integrable if and only if f is (PU)^{∗}-integrable and the two integral coincide.

Proof If f is (PU)^{∗}-measurable then by Proposition 6 it is μ−measurable and by completeness of measure also g is μ−measurable, then f −g = 0 a.e. and it is μ−measurable, μ−integrable and (PU)^{∗}-integrable with (P U)^{∗} ∫_{X}(f − g) = 0. So g = f − (f − g) is (PU)^{∗}-integrable.

Lemma 1 If f is a real μ−integrable function on X, A, B ∈ M, with A ⊂ B, and if c ∈ ℜ and ∫_{A} fdμ ≤ c ≤ ∫_{B} fdμ then there exists a μ−measurable set C such that A ⊂ C ⊂ B and ∫_{C} fdμ = c.

Proof Consider the σ−algebra D = {D ∈ M : D ⊂ B − A} and the signed measure α : D → ∫_{D} fdμ for D ∈ D.

By Liapounoff theorem (see [7]), the set {α(D) : D ∈ D} is a compact interval. So

α(∅) = 0 < c − ∫_{A} fdμ < ∫_{B−A} fdμ

and exists D_{1} ∈ D such that

∫_{D_{1}} fdμ = c − ∫_{A} fdμ

c = ∫_{A∪D_{1}} fdμ, A ⊂ A ∪ D_{1} ⊂ B.

Proposition 8 If f is a μ−measurable and PU^{∗}-integrable function on X, then for each ϵ > 0 there is a μ−measurable set E such that μ(X − E) < ϵ, f is μ−integrable on E and ∫_{E} fdμ = (P U)^{∗} ∫_{X} f.

Proof Suppose that f be not μ−integrable; set

E_{n} = {x ∈ X : n−1 ≤ f(x) < n}, F_{n} = {x ∈ X : −n ≤ f(x) < −n+1} n = 1, 2, 3, ...,

then

X = ⋃_{n=1}^{∞}(E_{n} ∪ F_{n}) = ⋃_{n=1}^{∞}(⋃_{i=1}^{n}(E_{i} ∪ F_{i})) = ⋃_{n=1}^{∞} H_{n},

where H_{n} = ⋃_{i=1}^{n}(E_{i} ∪ F_{i}) is an increasing sequence of measurable sets.

By a property of the measure, it results lim_{n→∞} μ(H_{n}) = μ(X) and for each ϵ > 0 there is n̄ ∈ N such that for n_{0} > n̄ it is

μ(X) − μ(H_{n_{0}}) = μ(X − H_{n_{0}}) < ϵ (∗)

f is bounded on H_{n_{0}} so it is μ−integrable on H_{n_{0}}.

Suppose that ∫_{H_{n_{0}}} fdμ < (P U^{∗}) ∫_{X} f; since f is not μ−integrable, then the series ∑_{n} ∫_{E_{n}} fdμ and ∑_{n} ∫_{F_{n}} fdμ are divergent to +∞ and to −∞ respectively. In fact, if ∑_{n} ∫_{E_{n}} fdμ = +∞ and ∑_{n} ∫_{F_{n}} fdμ > −∞, consider the functions

f_{1}(x) = f(x) if x ∈ ⋃_{n} E_{n} and f_{1}(x) = 0 elsewhere,

f_{2}(x) = f(x) if x ∈ ⋃_{n} F_{n} and f_{2}(x) = 0 elsewhere,

then f_{2}(x) is μ−integrable and hence (PU)^{∗}-integrable and f_{1}(x) = f(x) − f_{2}(x) is (PU)^{∗}-integrable, but it is also μ−integrable with integral +∞ and this is impossible. So for ϵ > 0 there exists K > n_{0} such that

∫_{H_{n_{0}}} fdμ + ∫_{E_{n_{0}+1}} fdμ + ..... + ∫_{E_{n_{0}+k}} fdμ > (P U)^{∗} ∫_{X} f

and set H = H_{n_{0}} ∪ E_{n_{0}+1} ∪ .... ∪ E_{n_{0}+k}, it results

∫_{H_{n_{0}}} fdμ < (P U)^{∗} ∫_{X} f < ∫_{H} fdμ.

By Lemma 1 there exists a μ−measurable set E with H_{n_{0}} ⊂ E ⊂ H such that ∫_{E} fdμ = (P U)^{∗} ∫_{X} f and by relation (*) we have:

μ(X − E) ≤ μ(X − H_{n_{0}}) < ϵ.

Lemma 2 If f is μ−measurable and there exists finite ∫_{X} fdμ, given ϵ > 0 there is a positive function δ on X such that

∑_{i} |(f(x_{i}) ∫_{X} θ_{i}dμ − ∫_{X} fθ_{i}dμ)| < ϵ

for each δ−fine (PU)^{∗}-partition P = {(θ_{i}, x_{i})} in X.

Proof See Proposition 3.1 in [5]

Proposition 9 A μ−measurable function f is (PU)^{∗}-integrable on X if and only if given ϵ > 0 there is a positive function δ on X and a μ−measurable set E such that μ(E^{C}) < ϵ, f is μ−integrable on E and | ∑_{i} fχ_{E^{C}} (x_{i}) ∫_{X} θ_{i}dμ| < ϵ for each δ−fine (PU)^{∗}-partition P = {(θ_{i}, x_{i})}. Moreover ∫_{E} fdμ = (P U)^{∗} ∫_{X} f. We have set E^{C} = X − E.

Proof If f is (PU)^{∗}-integrable, by previous Proposition, let ϵ > 0 there is E ∈ M such that μ(E^{C}) < ϵ, f is μ−integrable on E and ∫_{E} fdμ = (P U)^{∗} ∫_{X} f; so fχ_{E} is μ−integrable and hence (PU)^{∗}-integrable and

(P U)^{∗} ∫_{X} fχ_{E} = ∫_{X} fχ_{E}dμ = ∫_{E} fdμ = (P U)^{∗} ∫_{X} f.

By the (PU)^{∗}-integrability of f and fχ_{E}, at correspondence of ϵ > 0 there is a positive function δ on X such that for each δ−fine (PU)^{∗}-partition {(θ_{i}, x_{i})}, it results

| ∑_{i} f(x_{i}) ∫_{X} θ_{i}dμ − (P U)^{∗} ∫_{X} f| < \frac{ϵ}{2}

and

∑_{i} f(x_{i})χ_{E} ∫_{X} θ_{i}dμ − (P U)^{∗} ∫_{X} f| < \frac{ϵ}{2}.

So we have

| ∑_{i} f(x_{i})χ_{E^{C}} ∫_{X} θ_{i}dμ| = | ∑_{i} f(x_{i}) ∫_{X} θ_{i}dμ − ∑_{i} f(x_{i})χ_{E} ∫_{X} θ_{i}dμ| ≤

≤ | ∑_{i} f(x_{i}) ∫_{X} θ_{i}dμ − (P U)^{∗} ∫_{X} f| + | ∑_{i} fχ_{E}(x_{i}) ∫_{X} θ_{i}dμ − (P U)^{∗} ∫_{X} f| < ϵ.

Conversely, for ϵ > 0 let E be a μ−measurable and μ−integrable set with μ(E^{C}) < ϵ and let δ be a positive function on X such that | ∑_{i} fχ_{E}^{C}(x_{i}) ∫_{X} θ_{i}dμ| < \frac{ϵ}{2} for each δ−fine (PU)^{∗}-partition P in X.

By the μ−integrability of f on E , then also the function fχ_{E} is μ−integrable and, by lemma 2, there is a positive function δ_{1} on X such that

| ∑_{i} fχ_{E}(x_{i}) ∫_{X} θ_{i}dμ − ∫_{X} fχ_{E}dμ| < \frac{ϵ}{2}.

If δ̄(x) = min(δ(x), δ_{1}(x)) for each x ∈ X, then for each δ̄−fine (PU)^{∗}- partition P consider:

| ∑_{i} f(x_{i}) ∫_{X} θ_{i}dμ − ∫_{E} fdμ| ≤ | ∑_{i} fχ_{E}(x_{i}) ∫_{X} θ_{i}dμ − ∫_{E} fdμ|+

+| ∑_{i} fχ_{E}^{C}(x_{i}) ∫_{X} θ_{i}dμ| < \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ.

So f is (PU)^{∗}-integrable and (P U)^{∗} ∫_{X} f = ∫_{E} fdμ.

Proposition 10 If f and |f| are (PU)^{∗}-integrable then f is μ−integrable. Proof If f and |f| are (PU)^{∗}-integrable, consider the bounded sequence

f_{n} = |f| ∧ n for each n ∈ N it converges increasing to |f| and it is μ−integrable and

∫_{X} |f|dμ = lim_{n} ∫_{X} f_{n}dμ = lim_{n}(P U)^{∗} ∫_{X} f_{n} ≤ (P U)^{∗} ∫_{X} |f| < +∞.

So |f| and f are μ−integrable.

Proposition 11 If (f_{n})_{n} is an increasing sequence of (PU)^{∗}-integrable functions converging to f pointwise and lim_{n}(P U)^{∗} ∫_{X} f_{n} < ∞ then f is (PU)^{∗}- integrable and (P U)^{∗} ∫_{X} f = lim_{n}(P U)^{∗} ∫_{X} f_{n}.

Proof Consider the increasing sequence (f_{n} − f_{1})_{n} converging to f − f_{1}; since the functions (f_{n} − f_{1})_{n} are non negative, then by Proposition 10, they are μ−integrable and

lim_{n} ∫_{X}(f_{n} − f_{1})dμ = lim_{n}(P U)^{∗} ∫_{X}(f_{n} − f_{1}) =

= lim_{n}(P U)^{∗} ∫_{X} f_{n} − (P U)^{∗} ∫_{X} f_{1} < +∞.

So by the monotone theorem for the μ−integrable functions, the function (f − f_{1}) is μ−integrable and hence (PU)^{∗}-integrable. Therefore f = (f −f_{1})+f_{1} is (PU)^{∗}- integrable.

Proposition 12 If (f_{n})_{n} is a sequence of (PU)^{∗} integrable functions converging to f and such that there are two functions h and g (PU)^{∗}-integrable with h ≤ f_{n} ≤ g for each n ∈ N then f is (PU)^{∗}-integrable and (P U)^{∗} ∫_{X} f = lim_{n}(P U)^{∗} ∫_{X} f_{n}.

Proof Consider the sequence (f_{n} − h)_{n}; it is non negative and (PU)^{∗}- integrable, so it is μ−integrable and results:

0 ≤ (f_{n} − h) ≤ (g − h).

Since the function g−h is non negative and (PU)^{∗}-integrable, it is μ−integrable and by the dominate convergent theorem, the sequence of functions (f_{n} − h) converges to f − h which is a μ−integrable function and hence (PU)^{∗}-integrable. By the equality f = (f − h) + h it follows the (PU)^{∗}-integrability of f.

Proposition 13 If f is μ−measurable and exists finite ∫_{X} fdμ but ∫_{X} |f|dμ = +∞ then f is (PU)^{∗}-integrable and ∫_{X} fdμ = (P U)^{∗} ∫_{X} f. Proof If ϵ > 0, by lemma 2, there is positive function δ on X such that if

P = {(θ_{i}, x_{i})} is a (PU)^{∗}-partition in X, then we have:

ϵ > | ∑_{i}(f(x_{i}) ∫_{X} θ_{i}dμ − ∫_{X} fθ_{i}dμ)| = | ∑_{i} f(x_{i}) ∫_{X} θ_{i}dμ − ∑_{i} fθ_{i}dμ| =

= | ∑_{i}(f(x_{i}) ∫_{X} θ_{i}dμ − ∫_{X} fdμ)|.

An example of a function which is PU^{∗}-integrable but it is not PUintegrable.

Consider the space X = {0, 1}^{N}. Let ᾱ = α_{1}α_{2}...α_{k} be a finite string of 0 and 1; consider the set Aᾱ = [ᾱ]_{k} = {γ ∈ X : γ = ᾱβ, for some β ∈ X}, it is a clopen set (i.e. an open and closed set) with respect to the topology induced by the metric ρ so defined: if α, β ∈ X ρ(α, β) = \frac{1}{2^{n}} if α ≠ β and α_{1} = β_{1}, ...., α_{n} = β_{n}, α_{n+1} ≠ β_{n+1} ρ(α, α) = 0. With respect to this metric ρ, X = {0, 1}^{N} is a complete, separable and

compact metric space ( see [3]). Define on the family {Aᾱ} the following set function m:

m(Aᾱ) = \frac{1}{2^{k}}

and let m^{∗} be the outer measure induced by m on the family of all the subsets of X. If M is the σ−algebra of all the subsets of X m^{∗}−measurable in the Caratheodory sense, then the space (X, M, m^{∗}) satisfies the conditions α), β), γ) (see [3] and [5]).

Define on X the following real function

f(α) = \{ \begin{matrix}a_{1} if α_{1} = 0 \\ a_{2} if α_{1} = 1, α_{2} = 0 \\ a_{n} if α_{1}, α_{2}, ...α_{n−1} = 1, α_{n} = 0 \\ ... \\ . \end{matrix}

f(1111...111..) = 0

where α = (α_{1}, α_{2}, ...) ∈ \{0, 1\}^{N} and a_{n} = (−1)^{n} \frac{2^{n}}{n} .

Then, by Proposition 13, we have:

∫_{X} fdm = ∑_{n=1}^{∞} a_{n} \frac{1}{2^{n}} = ∑_{n=1}^{∞}(−1)^{n} \frac{1}{n} = (P U)^{∗} ∫_{X} f,

so f is PU^{∗}-integrable but |f| is not μ-integrable so it is not also (PU)-integrable.

References

[1] S.I.Ahmed and W.F.Pfeffer, A Riemann integral in a locally compact Hausdorff space, J.Australian Math.Soc.,(series A) 41 (1986) 115-137

[2] A.M.Bruckner, Differentiation of integrals, Americ. Math. Monthly, V.78(9) (1971)

[3] G.A.Edgar, Measure, topology and fractal geometry, Springer-Verlag,(1990)

[4] W.F.Pfeffer, T he Riemann approach to integration, Cambridge University Press (1993)

[5] G.Riccobono, A P U − Integral on an abstract metric space, Mathematica Bohemica 122 (1997) 83-95

[6] G.Riccobono, Convergence theorems for the P U − integral, Mathematica Bohemica 125 (2000) 77-86

[7] W.Rudin, F unctional Analysis, McGraw-Hill, N.York (1973)

[8] W.Rudin, P rinciples of Mathematical Analysis, McGraw-Hill, N.York (1976)