Step function (measure theory)

As a step function is called in the measure theory special real functions that the step functions are very similar. Jump functions can be found, for example, in the Lebesgue decomposition of functions or in the vicinity of Lebesgue-Stieltjes measures , where they characteristically form the distribution functions of purely atomic measures .

definition

A real function is called a step function if there is at most a countable set and a mapping ${\ displaystyle G}$ ${\ displaystyle A \ subset \ mathbb {R}}$

{\ displaystyle p \ colon A \ to (0, \ infty)}

there for that

{\ displaystyle \ sum _ {y \ in A \ cap [-n; n]} p (y) <\ infty}

applies to all and a representation as ${\ displaystyle n \ in \ mathbb {N}}$ ${\ displaystyle G}$

{\ displaystyle G (x) = {\ begin {cases} \ alpha - \ sum _ {y \ in A \ cap (x; 0]} p (y) & {\ text {for}} x <0 \\ \ alpha + \ sum _ {y \ in A \ cap (0; x]} p (y) & {\ text {for}} x \ geq 0 \ end {cases}}}

.

for one owns. ${\ displaystyle \ alpha \ in \ mathbb {R}}$

comment

The definition corresponds to the number of jump points and the function corresponds to the "weight" of the jump point, i.e. how much the function jumps upwards. The requirement for the weights ${\ displaystyle A}$ ${\ displaystyle p}$

{\ displaystyle \ sum _ {y \ in A \ cap [-n; n]} p (y) <\ infty}

ensures that there is not so much weight locally at one point that the function is unlimited up there. However, there can be an infinite number of weights in a small space as long as their overall contribution to function remains finite. It is also possible that a step function is unrestricted in the limit value . ${\ displaystyle \ pm \ infty}$

example

Graph of the Gaussian bracket

A typical example of a step function is the Gaussian bracket . It assigns the next lower whole number to each number, so it is given by

{\ displaystyle G (x): = \ max \ {k \ in \ mathbb {Z} \ mid k \ leq x \}}

The amount of jump points is one, so each jump point gets the weight ${\ displaystyle A = \ mathbb {Z}}$

{\ displaystyle p (k) = 1}

for everyone .

{\ displaystyle k \ in \ mathbb {Z}}

Demarcation

Jump functions are similar to, but generally different from, stair functions as well as simple functions .

Jump functions are always growing. This is not the case with staircase functions, nor with simple functions.
Jump functions can assume a countable number of values, step functions and simple functions only have a finite number of values.

literature

Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .