Measure, Integration & Real Analysis, 2021a

68 Chapter 2 Measures
For each interval I k of the form (b, c) with b < c and b, c ∈ R, define h on [b, c]
to be the linear function such that h(b) =g(b) and h(c) =g(c).
Define h(x) =g(x) for all x ∈ R for which h(x) has not been defined by the
previous two paragraphs. Then h : R → R is continuous and h| F = g.
The next result gives a slightly modified way to state Luzin’s Theorem. You can
think of this version as saying that the value of a Borel measurable function can be
changed on a set with small Lebesgue measure to produce a continuous function.
2.93 Luzin’s Theorem, second version
Suppose E ⊂ R and g : E → R is a Borel measurable function. Then for every
ε > 0, there exists a closed set F ⊂ E and a continuous function h : R → R such
that |E \ F| < ε and h| F = g| F .
Proof
Suppose ε > 0. Extend g to a function ˜g : R → R by defining
{
g(x) if x ∈ E,
˜g(x) =
0 if x ∈ R \ E.
By the first version of Luzin’s Theorem (2.91), there is a closed set C ⊂ R such
that |R \ C| < ε and ˜g| C is a continuous function on C. There exists a closed set
F ⊂ C ∩ E such that |(C ∩ E) \ F| < ε −|R \ C| (by 2.65). Thus
|E \ F| ≤ ∣ ∣ ( (C ∩ E) \ F ) ∪ (R \ C) ∣ ∣ ≤|(C ∩ E) \ F| + |R \ C| < ε.
Now ˜g| F is a continuous function on F. Also, ˜g| F = g| F (because F ⊂ E) . Use
2.92 to extend ˜g| F to a continuous function h : R → R.
The building at Moscow State University where the mathematics seminar organized
by Egorov and Luzin met. Both Egorov and Luzin had been students at Moscow State
University and then later became faculty members at the same institution. Luzin’s
PhD thesis advisor was Egorov.
CC-BY-SA A. Savin
Measure, Integration & Real Analysis, by Sheldon Axler