Measure, Integration & Real Analysis, 2021a

62 Chapter 2 Measures
2E
Convergence of Measurable Functions
Recall that a measurable space is a pair (X, S), where X is a set and S is a σ-algebra
on X. We defined a function f : X → R to be S-measurable if f −1 (B) ∈Sfor
every Borel set B ⊂ R. In Section 2B we proved some results about S-measurable
functions; this was before we had introduced the notion of a measure.
In this section, we return to study measurable functions, but now with an emphasis
on results that depend upon measures. The highlights of this section are the proofs of
Egorov’s Theorem and Luzin’s Theorem.
Pointwise and Uniform Convergence
We begin this section with some definitions that you probably saw in an earlier course.
2.82 Definition pointwise convergence; uniform convergence
Suppose X is a set, f 1 , f 2 ,... is a sequence of functions from X to R, and f is a
function from X to R.
• The sequence f 1 , f 2 ,... converges pointwise on X to f if
for each x ∈ X.
lim f k(x) = f (x)
k→∞
In other words, f 1 , f 2 ,... converges pointwise on X to f if for each x ∈ X
and every ε > 0, there exists n ∈ Z + such that | f k (x) − f (x)| < ε for all
integers k ≥ n.
• The sequence f 1 , f 2 ,... converges uniformly on X to f if for every ε > 0,
there exists n ∈ Z + such that | f k (x) − f (x)| < ε for all integers k ≥ n and
all x ∈ X.
2.83 Example a sequence converging pointwise but not uniformly
Suppose f k : [−1, 1] → R is the
function whose graph is shown here
and f : [−1, 1] → R is the function
defined by
{
1 if x ̸= 0,
f (x) =
2 if x = 0.
Then f 1 , f 2 ,...converges pointwise
on [−1, 1] to f but f 1 , f 2 ,... does
not converge uniformly on [−1, 1] to
f , as you should verify.
The graph of f k .
Measure, Integration & Real Analysis, by Sheldon Axler