The behavior at infinity of an integrable function
نویسندگان
چکیده
منابع مشابه
On the Behavior at Infinity of an Integrable Function
We denote by x a real variable and by n a positive integer variable. The reference measure on the real line R is the Lebesgue measure. In this note we will use only basic properties of the Lebesgue measure and integral on R. It is well known that the fact that a function tends to zero at infinity is a condition neither necessary nor sufficient for this function to be integrable. However, we hav...
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Given a density d de ned on the Borel subsets of [0;1); the limit at in nity in density of a function f : [0;1) ! R is zero if each of the sets ft : jf(t)j "g has zero density whenever " > 0: It is proved that every Lebesgue integrable function f : [0;1) ! R veri es this type of behavior at in nity with respect to a scale of densities including the usual one, d(A) = limr!1 m(A\[0;r)) r : The an...
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ژورنال
عنوان ژورنال: Expositiones Mathematicae
سال: 2012
ISSN: 0723-0869
DOI: 10.1016/j.exmath.2012.03.002