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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 $f$ on an interval $[a,b]$
Let $f:[a,b]\to \mathbb R$ be a measurable function. In this case the definition is following: For any set $E\subset[a,b]$ of measure zero ($|E|=0$), the image of this set, $f(E)$, also has measure zero. It was introduced by N.N. Luzin in 1915 (see [1]). The following assertions hold.
- A function $f\not\equiv \operatorname{const}$ on $[a,b]$ such that $f'(x)=0 $ almost-everywhere on $[a,b]$ (see for example Cantor ternary function) does not have the Luzin $\mathcal N$-property.
- If $f$ does not have the Luzin $\mathcal N$-property, then on $[a,b]$ there is a perfect set $P$ of measure zero such that $|f(P)|>0$.
- An absolutely continuous function has the Luzin $\mathcal N$-property.
- If $f$ has the Luzin $\mathcal N$-property and has bounded variation on $[a,b]$ (as well as being continuous on $[a,b]$), then $f$ is absolutely continuous on $[a,b]$ (the Banach–Zaretskii theorem).
- If $f$ does not decrease on $[a,b]$ and $f'$ is finite on $[a,b]$, then $f$ has the Luzin $\mathcal N$-property.
- In order that $f(E)$ be measurable for every measurable set $E\subset[a,b]$ it is necessary and sufficient that $f$ have the Luzin $\mathcal N$-property on $[a,b]$.
- A function $f$ that has the Luzin $\mathcal N$-property has a derivative $f'$ on the set for which any non-empty portion of it has positive measure.
- For any perfect nowhere-dense set $P\subset[a,b]$ there is a function $f$ having the Luzin $\mathcal N$-property on $[a,b]$ and such that $f'$ does not exist at any point of $P$.
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
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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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