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shows that D1 is a d-system. Moreover, D1 ⊃ C since A∩B ∈ C ⊂ D for every A ∈ C
by the fact that C is a π-system. So D1 must contain the smallest d-system containing
C, that is, D1 ⊃ D. In other words, A ∩ B ∈ D for every A ∈ D and B ∈ C.
Next, fix A ∈ D and let D2 = {B ∈ D : A ∩ B ∈ D}. We have just shown
that D2 ⊃ C. Moreover, by Lemma 24.4 again, D2 is a d-system. THus, D2 ⊃ D. In
other words, A ∩ B ∈ D for every A ∈ D and B ∈ D, that is, D is a π-system. This
completes the proof.
Exercises:
24.1 Partitions. A partition of E is a countable disjointed collection of subsets
whose union is E. It is called a finite partition if it has only finitely many
elements.
1. Let {A, B, C} be a partition of E. Describe the σ-algebra generated
by this partition.
2. Let C be a partition of E. Let E be the collection of all countable
unions of elements of C. Show that E is a σ-algebra. Show that, in
fact, E = σ(C).
Generally, if C is not a partition, the elements of σ(C) cannot be obtained
through such explicit constructions.
24.2 Let B and C be two collections of subsets of E. If B ⊂ C, then σ(B) ⊂
σ(C). If B ⊂ σ(C) ⊂ σ(B), then σ(B) = σ(C). Show these.
24.3 Borel σ-algebra on R. Show that B(R) is generated by the collection of
all open intervals. Hint: recall that every open subset of R is a countable
union of open intervals.
24.4 Continuation. Show that every interval of R is a Borel set. In particular,
(-∞, x), (-∞, x], (x, y], [x, y] are all Borel sets. Every singleton {x} is
a Borel set.
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MEASURE AND INTEGRATION
24.5 Show that B(R) is also generated by any one of the following:
1. the collection of all intervals of the form (x, ∞),
2. the collection of all intervals of the form (x, y],
3. the collection of all intervals of the form [x, y],
4. the collection of all intervals of the form (-∞, x],
5. the collection of all intervals of the form (x, ∞) with x rational.
25
Measurable Spaces and Functions
A measurable space is a pair (E, E) where E is a set and E is a σ-algebra on E. Then,
the elements of E are called measurable sets. When E is a metric space and E = B(E),
the Borel σ-algebra on E, the measurable sets are also called Borel sets.
Let (E, E) and F, F) be measurable spaces and let f be a mapping from E into F .
Then, f is said to be measurable relative to E and F if f-1(B) ∈ E for every B ∈ F
(these are the functions we wish to be able to integrate). If E and F are metric spaces
and E = B(E) and F = B(F ) and f : E → F is measurable relative to E and F, tthen
f is also called a Borel function.
Measurable Functions
The following proposition reduces the checks for measurability:
25.1 PROPOSITION. Let (E, E) and (F, F) be measurable spaces. In order for f :
E → F to be measurable relative to E and F, it is necessary and sufficient that
f-1(B) ∈ E for every B ∈ F0 for some collection F0 that generates F.
PROOF. Necessity part is trivial. To prove the sufficiency, let F0 ⊂ F be such that
σ(F0) = F and suppose that f-1(B) ∈ E for every B ∈ F0. We need to show that,
then,
F1 = {B ∈ F : f-1(B) ∈ E}
is equal to F. For this, it is sufficient to show that F1 is a σ-algebra, since F1 ⊃ F0
by hypothesis and F is the smallest σ-algebra containing F0. But checking that F1 is
a σ-algebra is easy in view of the relations given in Exercise 2.1.
25. MEASURABLE SPACES AND FUNCTIONS
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Borel Functions
Let E and F be metric spaces and let E and F be their respective Borel σ-algebras.
Let f : E → F . Since F is generated by the open subsets of F , in order for f to be a
Borel function, it is necessary and sufficient that f-1(B) ∈ E for every open subset B
of F ; this is an immediate corollary of the preceding proposition. In particular, if f is
continuous, then f-1(B) is open in E for every open B ⊂ F . Thus, every continuous
function f : E → F is Borel measurable. The converse is generally false.
Compositions of Functions
Let (E, E), (F, F), and (G, G) be measurable spaces. Let f : E → F and g : F → G.
Then, their composition g ◦ f : x → g(f(x)) is a mapping from E into G. The fol-
lowing proposition will be recalled by the phrase “measurable functions of measurable
functions are measurable”.
25.2 PROPOSITION. If f is measurable relative to E and F, and if g is measurable
relative to F and G, then g ◦ f is measurable relative to E and G.
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