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.)
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“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)
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