# On the decomposition theorems of Lebesgue and Jordan

### Real Analysis Exchange 23 (1997/8), no. 1, 313-324.

#### Mathematical reviews subject classification: 26A45, 26A46, 26A48.

## Abstract

The following decomposition theorem of Lebesgue is well known:

Theorem A : Lebesgue's decomposition theorem

(S. Saks, *Theory of the integral,* 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937).

If F is an additive function of bounded variation of an interval, the derivative F' is summable, and the function F is the sum of a singular additive function of an interval and of the indefinite integral of the derivative F'. Moreover, if the function F is non-negative, we have for every interval I_{o}

,

equality holding only in the case in which the function F is absolutely continuous on I_{o}.

In the first part of this paper, for the special case of a function defined on an interval [a,b], with bounded variation and satisfying Lusin's condition (N), Theorem A becomes

,

where s_F is the saltus function of F (clearly s_F is a singular function).

Moreover we obtain

(for the definition of S_{F} see Lemma 8).

The following decomposition theorem of Jordan is well known:

Theorem B.: Jordan's decomposition theorem

(I. P. Natanson, *Theory of functions of a real variable*, 2nd. rev. ed., Ungar, New York, 1961, p. 218)

A function F:[a,b] → R is VB if and only if it is representable as the difference of two increasing functions.

In fact from the proof of this theorem it follows that if F is VB on [a,b] then the functions V_{F}(x):= V(F,[a,x]) and G(x): = F(x) - V_{F}(x) are increasing.

The question is which properties of F will be preserved for V_{F} and G?

It is known that if F is left, right or bilaterally continuous at a point x ∈ [a,b] then so are V_{F} and G (see I. P. Natanson, *Theory of functions of a real variable*, 2nd. rev. ed., Ungar, New York, 1961, p. 223).

In the second part of this paper we show that if F satisfies Lusin's condition (N) then V_{F} and G also satisfy (N).