Properties of Measurable Functions
- The sum and product of two complex-valued measurable functions are measurable. So is the quotient, so long as there is no division by zero.
- The composition of measurable functions is measurable; i.e., if f: (X, Σ1) → (Y, Σ2) and g: (Y, Σ2) → (Z, Σ3) are measurable functions, then so is g(f(⋅)): (X, Σ1) → (Z, Σ3). But see the caveat regarding Lebesgue-measurable functions in the introduction.
- 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.
- The pointwise limit of a sequence of measurable functions is measurable; note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence. (This is correct when the counter domain of the elements of the sequence is a metric space. It is false in general; see pages 125 and 126 of.)
Read more about this topic: Measurable Function
Famous quotes containing the words properties of, properties and/or functions:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)