- Theorem
- Proof f is a step function f takes only positive values General case f takes finite values only Importance of the measure space
- See also
- References
In measure theory, the factorization lemma allows us to express a function f with another function T if f is measurable with respect to T. An application of this is regression analysis.
Theorem
Let 
be a function of a set 
in a measure space 
and let 
be a scalar function on . Then 
is measurable with respect to the σ-algebra 
generated by 
in if and only if there exists a measurable function 
such that 
, where 
denotes the Borel set of the real numbers. If only takes finite values, then 
also only takes finite values.
Proof
First, if , then f is 
measurable because it is the composition of a 
and of a 
measurable function. The proof of the converse falls into four parts: (1)f is a step function, (2)f is a positive function, (3) f is any scalar function, (4) f only takes finite values.
f is a step function
Suppose 
is a step function, i.e. 
and 
. As T is a measurable function, for all i, there exists 
such that 
. 
fulfils the requirements.
f takes only positive values
If f takes only positive values, it is the limit, for pointwise convergence, of a increasing sequence 
of step functions. For each of these, by (1), there exists 
such that 
. The function 
, which exists on the image of T for pointwise convergence because is monotonic, fulfils the requirements.
General case
We can decompose f in a positive part 
and a negative part 
. We can then find 
and 
such that 
and 
. The problem is that the difference 
is not defined on the set 
. Fortunately, 
because 
always implies 
We define 
and 
. 
fulfils the requirements.
f takes finite values only
If f takes finite values only, we will show that g also only takes finite values. Let 
. Then 
fulfils the requirements because 
.
Importance of the measure space
If the function is not scalar, but takes values in a different measurable space, such as 
with its trivial σ-algebra (the empty set, and the whole real line) instead of 
, then the lemma becomes false (as the restrictions on are much weaker).
See also
- Doob–Dynkin lemma
- Conditional expectation
References
- Heinz Bauer, Ed. (1992) Maß- und Integrationstheorie. Walter de Gruyter edition. 11.7 Faktorisierungslemma p. 71-72.
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