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measurable function
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(Definition)
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Let
and
be two measurable spaces. Then a function
is called a measurable function if:
where
.
In other words, the inverse image of every
-measurable set is
-measurable. The space of all measurable functions
is denoted as
Any measurable function into
, where
is the Borel sigma algebra of the real numbers
, is called a Borel measurable function.1 The space of all Borel measurable functions from a measurable space
is denoted by
.
Similarly, we write
for
, where
is the Borel sigma algebra of
, the set of extended real numbers.
Remark. If and are measurable functions, then so is
, for if is
-measurable, then is
-measurable, and
is
-measurable. But
, which implies that is a measurable function.
Example:
Footnotes
- 1
- More generally, a measurable function is called Borel measurable if the range space
is a topological space with
the sigma algebra generated by all open sets of .
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"measurable function" is owned by CWoo. [ full author list (3) | owner history (2) ]
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(view preamble)
Cross-references: measurable, characteristic function, subset, implies, extended real numbers, open sets, generated by, sigma algebra, topological space, range, real numbers, Borel sigma algebra, inverse image, function, measurable spaces
There are 47 references to this entry.
This is version 14 of measurable function, born on 2002-07-21, modified 2006-12-09.
Object id is 3176, canonical name is MeasurableFunctions.
Accessed 14576 times total.
Classification:
| AMS MSC: | 28A20 (Measure and integration :: Classical measure theory :: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence) |
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Pending Errata and Addenda
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