# User:Nikita2/sandbox

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*}

## Contents

#### $\mathcal N$-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

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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