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measurable function∗
CWoo†
2013-03-21 14:29:59
Let X, B(X) and Y, B(Y ) be two measurable spaces. Then a function
f : X → Y is called a measurable function if:
f −1 B(Y ) ⊆ B(X)
where f −1 B(Y ) = {f −1 (E) | E ∈ B(Y )}.
In other words, the inverse image of every B(Y )-measurable set is B(X)measurable. The space of all measurable functions f : X → Y is denoted as
M X, B(X) , Y, B(Y ) .
Any measurable function into (R, B(R)), where B(R) is the Borel sigma algebra
of the real numbers R, is called a Borel measurable function.1 The space of all
Borel measurable
functions from a measurable space (X, B(X)) is denoted by
L0 X, B(X) .
Similarly, we write L̄0 X, B(X) for M X, B(X)), (R̄, B(R̄) , where B(R̄)
is the Borel sigma algebra of R̄, the set of extended real numbers.
Remark. If f : X → Y and g : Y → Z are measurable functions, then so is
g ◦ f : X → Z, for
then g −1 (E)
if E is B(Z)-measurable, −1
is B(Y )-measurable,
−1 −1
and f
g (E) is B(X)-measurable. But f
g −1 (E) = (g ◦ f )−1 (E), which
implies that g ◦ f is a measurable function.
Example:
• Let E be a subset of a measurable space X. Then the characteristic
function χE is a measurable function if and only if E is measurable.
∗ hMeasurableFunctioni created: h2013-03-21i by: hCWooi version: h33176i Privacy
setting: h1i hDefinitioni h28A20i
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You can reuse this document or portions thereof only if you do so under terms that are
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1 More generally, a measurable function is called Borel measurable if the range space Y is
a topological space with B(Y ) the sigma algebra generated by all open sets of Y .
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