Review of Measurable Functions
Definition 1 (Measurable Functions) Let X and Y be nonempty sets and M and N
be σ–algebras of subsets of X and Y respectively.
(a) A function f : X → Y is said to be (M, N )–measurable if
E ∈ N =⇒ f −1 (E) ≡ x ∈ X f (x) ∈ E ∈ M
(b) A function f : X → IR or C is said to be M–measurable if
E ∈ BIR or BC =⇒ f −1 (E) ≡ x ∈ X f (x) ∈ E ∈ M
(c) A function f : IR → IR or C is said to be Borel–measurable if it is (BIR , BIR or BC )–
measurable and is said to be Lebesgue–measurable if it is (L, BIR or BC )–measurable, where
L is the set of all Lebesgue measurable subsets of IR.
(d) A function f : X → [0, ∞] is said to be M–measurable if
E ∈ BIR =⇒ f −1 (E) ∈ M
where IR = IR ∪ {−∞, ∞} is the extended real line and BIR is the σ–algebra
BIR = E ⊂ IR E ∩ IR ∈ BIR
Lemma 2 (Continuous Functions) If X and Y are metric spaces and f : X → Y is
continuous, then f is (BX , BY )–measurable.
Lemma 3 (Real Valued Functions) Let X be a nonempty set and M be a σ–algebra
of subsets of X. Let f : X → IR. Then
f is M–measurable ⇐⇒ f −1 (a, ∞) ∈ M ∀ a ∈ IR
⇐⇒ f −1 [a, ∞) ∈ M ∀ a ∈ IR
⇐⇒ f −1 (−∞, a) ∈ M ∀ a ∈ IR
⇐⇒ f −1 (−∞, a] ∈ M ∀ a ∈ IR
For f : X → [0, ∞] Then
f is M–measurable ⇐⇒ f −1 (a, ∞] ∈ M ∀ a > 0
⇐⇒ f −1 [a, ∞) ∈ M ∀ a ≥ 0
c Joel Feldman.
2008. All rights reserved.
November 14, 2008
Review of Measurable Functions
1
Theorem 4 (Measurable Function Toolbox) Let f, g : X → IR be M–measurable
and c ∈ IR, or let f, g : X → [0, ∞] be M–measurable and c > 0. Then
(a) f + c and cf are M–measurable.
(b) f + g is M–measurable.
(c) f g is M–measurable. (As usual, we use the convention that 0 × ∞ = 0.)
(d) max{f, g} is M–measurable. Here max{f, g} is the function from X to IR defined by
max{f, g}(x) = max{f (x), g(x)}.
(e) min{f, g} is M–measurable.
(f ) Let, for each n ∈ IN, the function fn : X → IR be M–measurable and let h : X → IR.
Or let, for each n ∈ IN, the function fn : X → [0, ∞] be M–measurable and let
h : X → [0, ∞]. If
h(x) = lim fn (x)
n→∞
or h(x) = inf fn (x)
n∈IN
or h(x) = sup fn (x)
n∈IN
or h(x) = lim inf fn (x)
n→∞
or h(x) = lim sup fn (x)
n→∞
for all x ∈ X, then h is M–measurable.
(g) If h : IR → IR is Borel measurable and g : X → IR is M–measurable, then h ◦ g is M
measurable. Here h ◦ g : X → IR is the function defined by h ◦ g(x) = h g(x) .
Definition 5 (Almost Everywhere) Let (X, M, µ) be a measure space and f, g, fn :
X → IR for all n ∈ IN. Then
(a) f = g a.e. if there is a set E ∈ M such that µ(E) = 0 and f (x) = g(x) for all x ∈
/ E.
(b) f = limn→∞ fn (x) a.e. if there is a set E ∈ M such that µ(E) = 0 and f =
limn→∞ fn (x) for all x ∈
/ E.
Lemma 6 (Almost Everywhere) Let (X, M, µ) be a measure space. The following
implications are true if and only if µ is complete.
(a) Let f, g : X → IR. If f is measurable and f = g µ–a.e., then g is measurable.
(b) Let f : X → IR and fn : X → IR for all n ∈ IN. If fn is measurable for all n ∈ IN
and {fn } converges µ–a.e. to f , then f is measurable.
c Joel Feldman.
2008. All rights reserved.
November 14, 2008
Review of Measurable Functions
2