In the mathematical field of mathematical analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) or Lusin's criterion states that an almost-everywhere finite function is measurable if and only if it is a continuous function on nearly all its domain. In the informal formulation of J. E. Littlewood, "every measurable function is nearly continuous".
For an interval [''a'', ''b''], let
f:[a,b] → C
be a measurable function. Then, for every ε > 0, there exists a compact E ⊆ [''a'', ''b''] such that f restricted to E is continuous and
\mu(E)>b-a-\varepsilon.
Note that E inherits the subspace topology from [''a'', ''b'']; continuity of f restricted to E is defined using this topology.
Also for any function f, defined on the interval [''a, b''] and almost-everywhere finite, if for any ε > 0 there is a function ϕ, continuous on [''a, b''], such that the measure of the set
\{x\in[a,b]:f(x) ≠ \phi(x)\}
is less than ε, then f is measurable.[1]
Let
(X,\Sigma,\mu)
f:X → Y
\varepsilon>0
A\in\Sigma
E
\mu(A\setminusE)<\varepsilon
f
E
A
Y=Rd
E
f\varepsilon:X → Rd
f
E
\supx\in|f\varepsilon(x)|\leq\supx\in|f(x)|
Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.
The proof of Lusin's theorem can be found in many classical books. Intuitively, one expects it as a consequence of Egorov's theorem and density of smooth functions. Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity.
1Q:[0,1]\to\{0,1\}
[0,1]
E.
Let
\{xn;n=1,2,...\}
Q
Gn=(x
n) | |
n+\varepsilon/2 |
infty | |
E:=[0,1]\setminuscup | |
n=1 |
Gn
Gn
E
1-2\varepsilon
Sources
Citations