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User:Nikita2/sandbox - Encyclopedia of Mathematics

# User:Nikita2/sandbox

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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 ). The following assertions hold.

1. 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.
2. 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 $|f(P)|>0$.
3. An absolutely continuous function has the Luzin $\mathcal N$-property.
4. 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).
5. If $f$ does not decrease on $[a,b]$ and $f'$ is finite on $[a,b]$, then $f$ has the Luzin $\mathcal N$-property.
6. 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 $[a,b]$.
7. 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.
8. 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.

How to Cite This Entry:
Nikita2/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nikita2/sandbox&oldid=29504