In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.
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Measurable function in the context of Layer cake representation
In mathematics, the layer cake representation of a non-negative, real-valued measurable function defined on a measure space is the formula
for all , where denotes the indicator function of a subset and denotes the (strict) super-level set:
