Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

Improper Integral Kurzweil-Henstock Integral in Riesz Spaces 85
We note that, if I k is an unbounded interval, then the element ξ k associated with I k is
necessarily +∞ or −∞ : otherwise γ ( ξ k ) should be a bounded interval and contain an unbounded
interval: contradiction.
Given any partition or decomposition Π = {( Ik, ξk)
: k = 1, …, p}
of [ A, B]
and a function
f : [ A, B]
→R , we call Riemann sum of f (and we write ∑ f ) the quantity
p
∑ | Ik | f( ξk)
,
(7.1)
k = 1
where in the sum in (7.1) only the terms for which I k is a bounded interval are included. This can
be required by simply postulating it or by defining the measure of an unbounded interval as +∞ , by
requiring f( +∞ ) = f(
−∞ ) = 0 and by means of the convention 0( ⋅ +∞ ) = 0(see
also [170], p. 65).
We now formulate our definition of Kurzweil-Henstock integral for functions defined on [ A, B]
.
Definition 7.2 We say that a function f : [ A, B]
→R is Kurzweil-Henstock integrable (in short
( KH ) -integrable) on [ A, B]
if there exists an element I ∈ R such that ∀ ε > 0 there exist a
function δ ∈Δ and a positive real number P such that
∑ f − I ≤ ε
(7.2)
Π
whenever Π= {( Ik, ξk)
: k = 1, … , p}
is a δ -fine partition of any bounded interval [ ab , ] with
[ ab , ] ⊃ [ AB , ] ∩[ − PP , ] and [ ab , ] ⊂ [ AB , ] . In this case we say that I is the ( KH ) -integral of f ,
and we denote the element I by the symbol ( ) B
KH ∫ f or more simply
A
B
∫ f . Later we will prove
A
that our integral is well-defined, that is such an I is uniquely determined.
We now prove the following characterization of ( KH ) -integrability:
Theorem 7.3 A function f : [ A, B]
→R is ( KH ) -integrable if and only if there exists J ∈ R such
that ∀ ε > 0 there exists a gauge γ such that
∑ f − J ≤ ε
(7.3)
Π
whenever Π= {( I , ξ ) : k = 1, … , p}
is a γ -fine partition of [ A, B]
, and in this case we have
∫
B
A
f J =
.
k k
Proof: We begin with the "only if" part. By hypothesis, ∀ ε > 0 there exist a function δ ∈Δ and a
positive real number P such that (7.2) holds. We now define on [ A, B]
a gauge γ in the following
way:
⎧(
ξ − δ( ξ) , ξ + δ( ξ)) if ξ∈
[ AB , ] ∩ R,
⎪
γξ ( ) = ⎨[
−∞,− P) if ξ=−∞
andA=−∞,
⎪
⎩(
P,+∞ ] if ξ =+∞ andB=+∞.
We observe that every γ -fine partition Π = {( Ik, ξk)
: k = 1, … , p}
of [ A, B]
is such that Ik ⊂ γ ( ξk)
∀ k = 1,
…, p.
In the case A =−∞, B = +∞ , the partition Π contains two unbounded intervals,
which we call J and K : of course, if inf J = −∞ and sup K = +∞ , then the ξ k ’s associated with J
Π