Suppose we are given a real-valued function f defined on a measurable set A. This value can be finite or infinite. When any real number a is chosen for this function f , if any of the following is a measurable set: the set of points in A that satisfy f ≧ a , the set of points in A that satisfy f < a , the set of points in A that satisfy f > a , or the set of points in A that satisfy f ≦ a , then f is said to be measurable on A , and f is called a measurable function on A. If f and g are measurable functions that take finite values, then the sum, difference, product, maximum, minimum, and quotient of the two (where f / g and g ≠ 0) are always measurable. Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information |
可測集合 Aの上で定義された実数値をとる関数 fが与えられているとする。この値は有限でも無限でもよいと考える。この関数 fに対して任意の実数 aをとるとき,f≧aを満足する Aの点の集合,f<aを満足する Aの点の集合,f>aを満足する Aの点の集合,f≦aを満足する Aの点の集合のうちどれかが可測集合になるならば,fは Aの上で可測であるといい,fを A上の可測関数という。 f,gが有限の値をとる可測関数ならば,両者の和,差,積,最大値,最小値,商 (ただし f/gで g≠0) も常に可測である。
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