What does the phrase "Lusin-type" mean? This might be a English question rather than a mathematical question, however I was wondering what the phrase "Lusin-type" refers to. I have seen a lot of theorems so-called "Lusin-type theorem" or "quantitative Lusin-type theorem". However I am not in the field of measure theory, I only know the Lusin's theorem, informally saying that "every measurable function is nearly continuous".
I feel the phrase "Lusin-type" means, except on a set of small measure, some property holds almost everywhere on the rest of domain.
 A: Indeed, the phrase "Lusin-type" seems to refer to results roughly of the form "For a small fee a rougher object is more regular." (typically "the rougher object" is a measurable function, "more regular" means continuity, and "the small fee" is that the statement holds off a set of small measure). Here is one example from the literature (from Alberti's paper "A Lusin type theorem for gradients".):
Theorem: Let $U\subseteq \mathbb{R}^d$ be an open subset of finite (and positive) Lebesgue measure. Then for any Borel measurable $f:U\to\mathbb{R}^d$, and for any $\epsilon\in\mathbb{R}_{>0}$, there is a $F_\epsilon\subseteq U$ closed in $U$, and a $C^1$ function $g:U\to\mathbb{R}$ vanishing on the boundary of $U$ such that

*

*$1-\epsilon < \dfrac{\text{Leb}_{\mathbb{R}^d}(F_\epsilon)}{\text{Leb}_{\mathbb{R}^d}(U)}$,

*$dg=f$ on $F_\epsilon$.

*$\exists C_d\in\mathbb{R}_{>0},\forall p\in[1,\infty]: |dg|_p \leq C_d \,\epsilon^{(1-p)/p} \,|f|_p$.

Thus in words, every measurable $1$-form on $U$ is exact (or equivalently every measurable vector field on $U$ is a gradient vector field) off a set of small measure.
Another related paper is Michael & Ziemer's "A Lusin Type Approximation of Sobolev Functions by Smooth Functions"; also see Ziemer's book Weakly Differentiable Functions, Section 3.10.
