Function for which the preimage of a measurable set is measurable
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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and in particular measure theory, a measurablefunction is a function between the underlying sets of two measurable spaces that preserves the structure...
Bochner-measurablefunction taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued...
weakly measurablefunction taking values in a Banach space is a function whose composition with any element of the dual space is a measurablefunction in...
Strong measurability has a number of different meanings, some of which are explained below. For a function f with values in a Banach space (or Fréchet...
defines integrals for a class of functions called measurablefunctions. A real-valued function f on E is measurable if the pre-image of every interval...
Class of mathematical sets Measurable function – Function for which the preimage of a measurable set is measurable Measure – Generalization of mass, length...
For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice...
random variable is defined as a measurablefunction from a probability measure space (called the sample space) to a measurable space. This allows consideration...
measurablefunctions on a measure space ( S , Σ , μ ) {\displaystyle (S,\Sigma ,\mu )} . Suppose that the sequence converges pointwise to a function f...
("pushing forward") a measure from one measurable space to another using a measurablefunction. Given measurable spaces ( X 1 , Σ 1 ) {\displaystyle (X_{1}...
same thing as "Lebesgue integrable" for measurablefunctions. The same thing goes for a complex-valued function. Let us define f + ( x ) = max ( ℜ f (...
interchanged. Likewise for sequences of non-negative pointwise-increasing (measurable) functions 0 ≤ f 1 ( x ) ≤ f 2 ( x ) ≤ ⋯ {\displaystyle 0\leq f_{1}(x)\leq...
applications, for example in probability theory. Definition 1. A measurablefunction L : (0, +∞) → (0, +∞) is called slowly varying (at infinity) if for...
) [ y − x ] {\displaystyle f(y)\leq f(x)+f'(x)[y-x]} A Lebesgue measurablefunction on an interval C is concave if and only if it is midpoint concave...
real-valued Lebesgue measurablefunction that is midpoint-convex is convex: this is a theorem of Sierpiński. In particular, a continuous function that is midpoint...
\Omega \to \mathbb {R} ^{+}} is a non-negative measurablefunction. In this context, the weight function w ( x ) {\displaystyle w(x)} is sometimes referred...
{R} ^{n}} and f : Ω → C {\displaystyle \mathbb {C} } be a Lebesgue measurablefunction. If f on Ω is such that ∫ K | f | d x < + ∞ , {\displaystyle \int...
{\displaystyle \{s\in S:f(s)\neq g(s)\}} is measurable and has measure zero. Similarly, a measurablefunction f {\displaystyle f} (and its absolute value)...
called Brownian motion) in the form of a measurable map from the unit interval to the space of continuous functions. The theory of standard probability spaces...
Measurablefunction: the preimage of each measurable set is measurable. Borel function: the preimage of each Borel set is a Borel set. Baire function...
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