Measure, Integration & Real Analysis, 2021a

Section 2E Convergence of Measurable Functions 65
Approximation by Simple Functions
2.88 Definition simple function
A function is called simple if it takes on only finitely many values.
Suppose (X, S) is a measurable space, f : X → R is a simple function, and
c 1 ,...,c n are the distinct nonzero values of f . Then
f = c 1 χ
E1
+ ···+ c n χ
En
,
where E k = f −1 ({c k }). Thus this function f is an S-measurable function if and
only if E 1 ,...,E n ∈S(as you should verify).
2.89 approximation by simple functions
Suppose (X, S) is a measurable space and f : X → [−∞, ∞] is S-measurable.
Then there exists a sequence f 1 , f 2 ,...of functions from X to R such that
(a) each f k is a simple S-measurable function;
(b) | f k (x)| ≤|f k+1 (x)| ≤|f (x)| for all k ∈ Z + and all x ∈ X;
(c)
(d)
lim
k→∞
f k (x) = f (x) for every x ∈ X;
f 1 , f 2 ,...converges uniformly on X to f if f is bounded.
Proof The idea of the proof is that for each k ∈ Z + and n ∈ Z, the interval
[n, n + 1) is divided into 2 k equally sized half-open subintervals. If f (x) ∈ [0, k],
we define f k (x) to be the left endpoint of the subinterval into which f (x) falls; if
f (x) ∈ [−k,0), we define f k (x) to be the right endpoint of the subinterval into
which f (x) falls; and if | f (x)| > k, we define f k (x) to be ±k. Specifically, let
⎧
m
if 0 ≤ f (x) ≤ k and m ∈ Z is such that f (x) ∈ [ m
, m+1 )
2 k 2 k 2 ,
k
⎪⎨ m+1
if − k ≤ f (x) < 0 and m ∈ Z is such that f (x) ∈ [ m
, m+1 )
2
f k (x) =
k 2 k 2 ,
k
k if f (x) > k,
⎪⎩
−k if f (x) < −k.
Each f −1( [ m 2 k , m+1
2 k ) ) ∈Sbecause f is an S-measurable function. Thus each f k
is an S-measurable simple function; in other words, (a) holds.
Also, (b) holds because of how we have defined f k .
The definition of f k implies that
2.90 | f k (x) − f (x)| ≤ 1 2 k for all x ∈ X such that f (x) ∈ [−k, k].
Thus we see that (c) holds.
Finally, 2.90 shows that (d) holds.
Measure, Integration & Real Analysis, by Sheldon Axler