Simple function

A simple function on a unit E is a numerical Fonction which takes only one finished number of values i.e the unit F (E) is finished.

Characterization of the simple functions

The simple functions are the linear combinations of indicatrixes. Indeed, let us suppose that $f=\; \backslash \; sum\_\; \{k=1\}\; ^N\; \backslash \; lambda\_k\; 1\_\; \{A\_k\}$, one has $f\; (E)\; \backslash \; subset\; \backslash \; \{f=\; \backslash \; sum\_\; \{k=1\}\; ^N1\_\; \{A\_k\}\; \backslash \; epsilon\_k\; ~~|\; \backslash \; epsilon\_k\; \backslash \; in\; \backslash \; \{0,\; 1\; \backslash \}\; ^\; \{\backslash \; \{1,\dots ,\; NR\; \backslash \}\}\; \backslash \}$

Let us suppose that F (E) is finished and note $a\_1,\dots \; a\_N$ the nonnull values of F, one poses $A\_j=f^\; \{-\; 1\}\; (a\_j)$ then the units $A\_1,\dots .,\; A\_N$ are 2 to 2 disjoined and there is $f=\; \backslash \; sum\_\; \{k=1\}\; ^N\; \backslash \; lambda\_k\; 1\_\; \{A\_k\}$

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