Topological ring

An extension of the concept of the integral, proposed by P. Daniell [1]. The construction scheme of this integral, which is known as the Daniell scheme, is an extension of the integral originally defined for a certain set of functions — the so-called elementary functions — to a wider class of functions. While retaining the way of extending but by changing the content of the initial set of elementary functions, it is possible to arrive at different extensions of the concept of the integral. In this scheme the concept of an elementary integral is axiomatically defined, unlike in Lebesgue's scheme (cf. Lebesgue integral) in which the concept of a measure is axiomatic.

Let $ X $ be an arbitrary set and let $ L _ {0} $ be a certain set of real bounded functions defined on $ X $; these functions are called elementary. It is assumed that $ L _ {0} $ is a vector lattice, i.e. from $ f, g \in L _ {0} $ and $ \alpha , \beta \in \mathbf R $ it follows that $ \alpha f+ \beta g \in L _ {0} $, and

$$ f, g \in L _ {0} \ \textrm{ implies } \ \sup ( f, g) , \inf ( f, g) \in L _ {0} . $$

Let $ I $ be a real functional defined on $ L _ {0} $ such that

1) $ I( \alpha f + \beta g) = \alpha I ( f ) + \beta I ( g) $( linearity);

2) $ f \geq 0 \Rightarrow I( f ) \geq 0 $( non-negativity);

3) if $ f _ {n} ( x) \downarrow 0 $ for all $ x $, then $ I ( f _ {n} ) \rightarrow 0 $( continuity with respect to monotone convergence).

Such a functional is known as an integral over elementary functions or an elementary integral. A set $ M \subset X $ is said to be a set of measure zero if for each $ \epsilon > 0 $ there exists a non-decreasing sequence $ \{ g _ {n} \} \subset L _ {0} $ such that $ \sup _ {n} g _ {n} ( x) \geq \chi _ {M} ( x) $ for all $ x $, and

$$ \sup I ( g _ {n} ) \leq \epsilon . $$

Here, $ \chi _ {M} $ denotes the characteristic function of $ M $.

A function $ f $ defined on $ X $ belongs to the class $ L ^ {+} $ if there exists a sequence $ \{ f _ {n} \} \in L _ {0} $ such that $ f _ {n} ( x) \uparrow f( x) $[[ Almost-everywhere|almost-everywhere]] and $ I ( f _ {n} ) \leq c < + \infty $. The number

$$ I( f ) = \lim\limits _ { n } I( f _ {n} ) $$

is said to be the integral of $ f $. The integral $ I( f ) $ does not depend on the choice of the particular approximating sequence $ \{ f _ {n} \} $.

The class $ L $ is the set of functions $ f $ which are defined on $ X $ and which are representable in the form $ f= f _ {1} - f _ {2} $, where $ f _ {1} , f _ {2} \in L ^ {+} $. Functions of the class $ L $ are called summable, while the number

$$ I( f ) = I( f _ {1} ) - I( f _ {2} ) $$

is known as the Daniell integral of the function $ f $. The class $ L $ is a vector lattice of finite functions (considered up to sets of measure zero) which is closed with respect to almost-everywhere convergence, with finite integrals, while the Daniell integral of summable functions has the properties of linearity, non-negativity, continuity with respect to almost-everywhere convergence of majorable summable functions (Lebesgue's theorem on passing to the limit under the integral sign), and also several other natural properties of the integral.

If $ X= [ a, b] $ and $ L _ {0} $ is the set of the step functions

$$ f( x) = c _ {i} , a _ {i} \leq x < b _ {i} ,\ \cup _ { i= } 1 ^ { n } [ a _ {i} , b _ {i} ) = [ a, b), b _ {i} = a _ {i+} 1 , $$

the Daniell integral becomes identical with the Lebesgue integral on the summable functions on $ [ a, b] $. The Daniell scheme may be used to construct the integral of functions with values in a $ \sigma $- complete lattice.

References

[1]	P. Daniell, "A general form of integral" Ann. of Math , 19 (1917) pp. 279–294
[2]	G.E. Shilov, B.L. Gurevich, "Integral, measure, and derivative: a unified approach" , Dover, reprint (1977) (Translated from Russian)
[3]	L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953)

Comments

Property 3) to be satisfied by the non-negative linear functional $ I $ above (i.e. $ I ( f _ {n} ) \rightarrow 0 $ as $ f _ {n} ( x) \downarrow 0 $ for all $ x $) is called the Denjoy condition, and is a very important requirement.

In the article above, functions in $ L ^ {+} $( in $ L $) differing on a set of measure zero are tacitly identified; the equivalence classes thus obtained are also called functions (with some abuse of language), as is usually done in measure theory. The statement that $ L $ be a vector lattice is thus to be understood as: the set of equivalence classes in $ L $ forms a vector lattice.

If the vector lattice $ L _ {0} $ has the property

$$ f \in L _ {0} \textrm{ implies } \inf ( 1, f ) \in L _ {0} , $$

then there is on the $ \sigma $- field generated by $ L _ {0} $ on $ X $ a unique $ \sigma $- finite $ \sigma $- additive measure $ \mu $ such that $ L $ is $ L _ {1} ( \mu ) $, and $ I ( f ) $ is $ \int f d \mu $ for $ f \in L $( see [3]). Actually, the Daniell integral is often used for constructing measures in functional analysis.

References

[a1]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 199–206; 334
[a2]	B. Szökefalvi-Nagy, "Real functions and orthogonal expansions" , Oxford Univ. Press (1965)