A staged function is a measurable Fonction whose image is finished. These functions play a big role in Théorie of integration within the meaning of Lebesgue. It is about a generalization of the functions in staircase used in theory of the Intégrale of Riemann.

Index property

A function in staircase is a finished linear Combinaison indicating functions of measurable units. In other words, are ( X , Σ) a measurable space, has ₁,…, has _N ∈ Σ a continuation finished measurable units, and has ₁,…, has _N a finished succession of real numbers or complex. A staged function is a function of the form:

$f\; (X)\; =\; \backslash \; sum\_\; \{k=1\}\; ^n\; a\_k\; \{\backslash \; mathbf\; 1\}\; \_\; \{A\_k\}\; (X).$

Together functions in staircase

Structure

It rises from the definition that the sum, the product of two functions in staircase, the product of a function in staircase by a complex is a function in staircase. The whole of the functions in staircase thus constitutes a C - commutative algebra.

Density

; Theorem:

the whole of the positive functions in staircase is dense in the whole of the positive measurable functions.

This theorem is equivalent to:

Any function measurable is the simple limit of functions in staircase.

; Demonstration

Is $f$ a positive function definite on a measurable space $(\backslash \; Omega,\; \{\backslash \; mathcal\; F\},\; \backslash \; driven)$. For all $n\; \backslash \; in\; \backslash \; mathbb\; N$, one shares the image of $f$ in $2^\; \{2n\}\; +1$ intervals length $2^\; \{-\; N\}$. One poses $I\_\; \{N,\; K\}\; =\; for$ k=1,2,\; \backslash \; ldots,\; 2^\; \{2n\}$and$ I\_\; \{N,\; 2^\; \{2n\}\; +1\}\; =\; [2^n,\; +\; \backslash \; infty\; [$.\; One\; defines\; the\; measurable\; units$ A\_\; \{N,\; K\}\; =f^\; \{-\; 1\}\; (I\_\; \{N,\; K\})$for$ k=1,2,\; \backslash \; ldots,\; 2^\; \{2n\}$.\; Then\; the\; increasing\; continuation\; of\; functions$ f\_n=\; \backslash \; sum\_\; \{k=1\}\; ^\; \{2^\; \{2n\}\; +1\}\; \backslash \; frac\; \{k-1\}\; \{2^n\}\; \{\backslash \; mathbf\; 1\}\; \_\; \{A\_\; \{N,\; K\}\}$converges\; simply\; towards$ f$when$ n$tends\; towards$ +\; \backslash \; infty$.\; ;\; Notice$

If F is limited, the action pursuant taken in the demonstration above converges uniformly.

Integration of a function in staircase

That is to say a measurement μ defined on ( X , Σ), the Integral of Lebesgue of F compared to μ is

$\backslash \; sum\_\; \{k=1\}\; ^na\_k\; \backslash \; driven\; (A\_k),$

when each term of the sum above is finished.

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