Measure, Integration & Real Analysis, 2021a

32 Chapter 2 Measures
The definition of an S-measurable
function requires the inverse image
of every Borel subset of R to be in
S. The next result shows that to verify
that a function is S-measurable,
we can check the inverse images of
a much smaller collection of subsets
of R.
Note that if f : X → R is a function and
a ∈ R, then
f −1( (a, ∞) ) = {x ∈ X : f (x) > a}.
2.39 condition for measurable function
Suppose (X, S) is a measurable space and f : X → R is a function such that
f −1( (a, ∞) ) ∈S
for all a ∈ R. Then f is an S-measurable function.
Proof
Let
T = {A ⊂ R : f −1 (A) ∈S}.
We want to show that every Borel subset of R is in T . To do this, we will first show
that T is a σ-algebra on R.
Certainly ∅ ∈T, because f −1 (∅) =∅ ∈S.
If A ∈T, then f −1 (A) ∈S; hence
f −1 (R \ A) =X \ f −1 (A) ∈S
by 2.33(a), and thus R \ A ∈T. In other words, T is closed under complementation.
If A 1 , A 2 ,...∈T, then f −1 (A 1 ), f −1 (A 2 ),...∈S; hence
f −1( ∞ ⋃
k=1
A k
)
=
∞⋃
k=1
f −1 (A k ) ∈S
by 2.33(b), and thus ⋃ ∞
k=1
A k ∈T. In other words, T is closed under countable
unions. Thus T is a σ-algebra on R.
By hypothesis, T contains {(a, ∞) : a ∈ R}. Because T is closed under
complementation, T also contains {(−∞, b] : b ∈ R}. Because the σ-algebra T is
closed under finite intersections (by 2.25), we see that T contains {(a, b] : a, b ∈ R}.
Because (a, b) = ⋃ ∞
k=1
(a, b − 1 k ] and (−∞, b) =⋃ ∞
k=1
(−k, b − 1 k
] and T is closed
under countable unions, we can conclude that T contains every open subset of R.
Thus the σ-algebra T contains the smallest σ-algebra on R that contains all open
subsets of R. In other words, T contains every Borel subset of R. Thus f is an
S-measurable function.
In the result above, we could replace the collection of sets {(a, ∞) : a ∈ R}
by any collection of subsets of R such that the smallest σ-algebra containing that
collection contains the Borel subsets of R. For specific examples of such collections
of subsets of R, see Exercises 3–6.
Measure, Integration & Real Analysis, by Sheldon Axler