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A mapping $\varphi:D\to D'$ possesses Luzin's $\mathcal N$-property if the image of every set of measure zero is a set of measure zero. A mapping $\varphi$ possesses Luzin's $\mathcal N{}^{-1}$-property if the preimage of every set of measure zero is a set of measure zero.
Briefly \begin{equation*} \mathcal N\text{-property:}\quad \Sigma\subset D, |\Sigma| = 0 \Rightarrow |\varphi(\Sigma)|=0, \end{equation*} \begin{equation*} \mathcal N{}^{-1}\text{-property:} \quad M \subset D, |M| = 0 \Rightarrow |\varphi^{-1}(M)|=0. \end{equation*}
$\mathcal N$-property of a function $f$, continuous on an interval $[a,b]$
For any set of measure
, the image of this set,
, also has measure zero. It was introduced by N.N. Luzin in 1915 (see [1]). The following assertions hold.
1) A function on
such that
almost-everywhere on
does not have the Luzin
-property.
2) If does not have the Luzin
-property, then on
there is a perfect set
of measure zero such that
.
3) An absolutely continuous function has the Luzin -property.
4) If has the Luzin
-property and has bounded variation on
(as well as being continuous on
), then
is absolutely continuous on
(the Banach–Zaretskii theorem).
5) If does not decrease on
and
is finite on
, then
has the Luzin
-property.
6) In order that be measurable for every measurable set
it is necessary and sufficient that
have the Luzin
-property on
.
7) A function that has the Luzin
-property has a derivative
on the set for which any non-empty portion of it has positive measure.
8) For any perfect nowhere-dense set there is a function
having the Luzin
-property on
and such that
does not exist at any point of
.
The concept of Luzin's -property can be generalized to functions of several variables and functions of a more general nature, defined on measure spaces.
References
[1] | N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1915) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212) |
Comments
There is another property intimately related to the Luzin -property. A function
continuous on an interval
has the Banach
-property if for all Lebesgue-measurable sets
and all
is a
such that
![]() |
This is clearly stronger than the -property. S. Banach proved that a function
has the
-property (respectively, the
-property) if and only if (respectively, only if — see below for the missing "if" ) the inverse image
is finite (respectively, is at most countable) for almost-all
in
. For classical results on the
- and
-properties, see [a3].
Recently a powerful extension of these results has been given by G. Mokobodzki (cf. [a1], [a2]), allowing one to prove deep results in potential theory. Let and
be two compact metrizable spaces,
being equipped with a probability measure
. Let
be a Borel subset of
and, for any Borel subset
of
, define the subset
of
by
(if
is the graph of a mapping
, then
). The set
is said to have the property (N) (respectively, the property (S)) if there exists a measure
on
(here depending on
) such that for all
,
![]() |
(respectively, for all there is a
such that for all
one has
![]() |
Now has the property (N) (respectively, the property (S)) if and only if the section
of
is at most countable (respectively, is finite) for almost-all
.
References
[a1] | C. Dellacherie, D. Feyel, G. Mokobodzki, "Intégrales de capacités fortement sous-additives" , Sem. Probab. Strasbourg XVI , Lect. notes in math. , 920 , Springer (1982) pp. 8–28 MR0658670 Zbl 0496.60076 |
[a2] | A. Louveau, "Minceur et continuité séquentielle des sous-mesures analytiques fortement sous-additives" , Sem. Initiation à l'Analyse , 66 , Univ. P. et M. Curie (1983–1984) Zbl 0587.28003 |
[a3] | S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05 |
[a4] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202 |
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