Measure, Integration & Real Analysis, 2021a

30 Chapter 2 Measures
Inverse images have good algebraic properties, as is shown in the next two results.
2.33 algebra of inverse images
Suppose f : X → Y is a function. Then
(a) f −1 (Y \ A) =X \ f −1 (A) for every A ⊂ Y;
(b) f −1 ( ⋃ A∈A A) = ⋃ A∈A f −1 (A) for every set A of subsets of Y;
(c) f −1 ( ⋂ A∈A A) = ⋂ A∈A f −1 (A) for every set A of subsets of Y.
Proof
Suppose A ⊂ Y. Forx ∈ X we have
x ∈ f −1 (Y \ A) ⇐⇒ f (x) ∈ Y \ A
⇐⇒ f (x) /∈ A
⇐⇒ x /∈ f −1 (A)
⇐⇒ x ∈ X \ f −1 (A).
Thus f −1 (Y \ A) =X \ f −1 (A), which proves (a).
To prove (b), suppose A is a set of subsets of Y. Then
x ∈ f −1 ( ⋃
A) ⇐⇒ f (x) ∈ ⋃
A
A∈A
A∈A
⇐⇒ f (x) ∈ A for some A ∈A
⇐⇒ x ∈ f −1 (A) for some A ∈A
⇐⇒ x ∈ ⋃
f −1 (A).
A∈A
Thus f −1 ( ⋃ A∈A A) = ⋃ A∈A f −1 (A), which proves (b).
Part (c) is proved in the same fashion as (b), with unions replaced by intersections
and for some replaced by for every.
2.34 inverse image of a composition
Suppose f : X → Y and g : Y → W are functions. Then
(g ◦ f ) −1 (A) = f −1( g −1 (A) )
for every A ⊂ W.
Proof Suppose A ⊂ W. Forx ∈ X we have
x ∈ (g ◦ f ) −1 (A) ⇐⇒ (g ◦ f )(x) ∈ A ⇐⇒ g ( f (x) ) ∈ A
⇐⇒ f (x) ∈ g −1 (A)
⇐⇒ x ∈ f −1( g −1 (A) ) .
Thus (g ◦ f ) −1 (A) = f −1( g −1 (A) ) .
Measure, Integration & Real Analysis, by Sheldon Axler