Measurable function

A measurable function is defined in mathematics as a function of a measurement space into another metric space, in which the prototype is any measurable subset of a measurable subset of. Such a function is also called - measurable.

Special cases

  • If the target range of the function is a normed or metric space, we equip this usually with the Borel σ - algebra generated by this and mentioned the algebra no further, but simply speaks of measurability.
  • Is the domain of the function of, how to equip these usually with the σ - algebra of Lebesgue measurable sets from.
  • A function satisfies both of the above conditions. It is according to measurable if the inverse image of Borel sets is a Lebesgue measurable set.

Classification

The notion of measurability is motivated by the definition of the integration of Henri Lebesgue: For the Lebesgue integration of a function with respect to the Lebesgue measure sets of the form a measure must be assigned. Examples of functions, for which this is not possible, are indicator functions Vitali quantities. The definition of the Lebesgue integration for arbitrary measure spaces leads to the above definition of measurable function.

The concept of measurable function has parallels to the definition of continuous function. A function between topological spaces and continuous if the inverse images of open sets of open sets of turn are. The heat generated by the open sets σ - algebra is the Borel σ - algebra. A continuous function that is measurable with respect to the Borel σ - algebras of short and borel measurable. A certain converse of this statement is the set of Lusin.

Measurable functions play an important role in probability theory as random variables.

Properties

Indicator functions of measurable quantities and linear combinations of such functions (so-called simple functions ) are examples of measurable functions from a measure space to the real numbers, equipped with the Borel σ - algebra.

The concatenation of measurable functions is again a measurable function, more precisely, f - g and measurable - measurable, then g o f - measurable.

A (real) Lebesgue - Borel - measurable function is not necessarily Borel Borel measurable. Also a Lebesgue - Borel - measurable function is not necessarily Lebesgue - Lebesgue measurable. The concatenation of two Lebesgue measurable Borel functions is therefore not necessarily turn Lebesgue - Borel - measurable.

Strong measurability

If a function in a metric space pointwise limit of elementary functions, ie measurable functions with finite image, it is called " strongly measurable".

  • Each measurable function with separablem image is strongly measurable.
  • Each strongly measurable function is measurable.

Strong measurability and measurability only differ from each other when the target space is non- separable. This is the case for example in the definition of generalized integrals as the Bochner integral.

Examination of the measurability of generating systems

If the σ - algebras generated by the quantity and systems; So, it suffices to prove the measurability of f for all.

Thus applies for a mapping f from a measurement facility according to that f is measurable if one of the systems of sets

  • ,
  • ,
  • ,

Is in ( if it is taken as σ - algebra on the Borel σ - algebra). Here, etc. as an abbreviation for to understand. It would also suffice if the a only runs through all rational numbers.

Demarcation

A subset of a measurement space is called measurable if it is an element of the σ - algebra of the measurement space and its therefore a measure can be assigned.

Lusin's theorem Bochner-Integral
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