Editing Measurable function (section)

== Properties of measurable functions ==

* The sum and product of two complex-valued measurable functions are measurable.<ref name="folland">{{cite book|last=Folland|first=Gerald B.|title=Real Analysis: Modern Techniques and their Applications|year=1999|publisher=Wiley|isbn=0-471-31716-0}}</ref> So is the quotient, so long as there is no division by zero.<ref name="strichartz" />
* If <math>f : (X,\Sigma_1) \to (Y,\Sigma_2)</math> and <math>g:(Y,\Sigma_2) \to (Z,\Sigma_3)</math>  are measurable functions, then so is their composition <math>g\circ f:(X,\Sigma_1) \to (Z,\Sigma_3).</math><ref name="strichartz" />
* If <math>f : (X,\Sigma_1) \to (Y,\Sigma_2)</math> and <math>g:(Y,\Sigma_3) \to (Z,\Sigma_4)</math> are measurable functions, their composition <math>g\circ f: X\to Z</math> need not be <math>(\Sigma_1,\Sigma_4)</math>-measurable unless <math>\Sigma_3 \subseteq \Sigma_2.</math> Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable.
* The (pointwise) [[supremum]], [[infimum]], [[limit superior]], and [[limit inferior]] of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.<ref name="strichartz" /><ref name="royden">{{cite book|last=Royden|first=H. L.|title=Real Analysis|year=1988|publisher=Prentice Hall|isbn=0-02-404151-3}}</ref>
*The [[pointwise]] limit of a sequence of measurable functions <math>f_n: X \to Y</math> is measurable, where <math>Y</math> is a metric space (endowed with the Borel algebra). This is not true in general if <math>Y</math> is non-metrizable. The corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence.<ref name="dudley">{{cite book|last=Dudley|first=R. M.|title=Real Analysis and Probability|year=2002|edition=2|publisher=Cambridge University Press|isbn=0-521-00754-2}}</ref><ref name="aliprantis">{{cite book|last1=Aliprantis|first1=Charalambos D.|last2=Border|first2=Kim C.|title=Infinite Dimensional Analysis, A Hitchhiker's Guide|year=2006|edition=3|publisher=Springer|isbn=978-3-540-29587-7}}</ref>