let f n : R → R be a nonnegative sequence of Borel measurable functions which converge to a function f, and the integral of all f n and of f are all 1. Then does the integral of | f n − f | goes to 0 as n → ∞ ??

Question

let       f    n    :      R    →      R   be a nonnegative sequence of Borel measurable functions which converge to a function   f, and the integral of all       f    n   and of   f are all 1. Then does the integral of       |        f    n    −  f      |   goes to 0 as   n  →  ∞  ??

Gornil2 · Accepted Answer

General Lebesgue Dominated Convergence Theorem:-Let   {      g          n        } be a sequence of integrable function which converge almost surely to an integrable function g. If   {      f          n        } be a sequence of functions such that       |        f          n            |    ≤      g          n       and   {      f          n        } converges to   f almost surely. If       ∫          X        g    d  μ  =      lim          n      →      ∞            ∫          X            g          n          d  μ then       ∫          X        f    d  μ  =      lim          n      →      ∞            ∫          X            f          n          d  μMore generally:- If   {      f          n        } is a sequence of integrable functions from a measure space   (  X  ,      F    ,  μ and       f          n        →  f almost surely and f  f is integrable. Then       ∫          X            |        f          n        −  f      |      d  μ  →  0 iff       lim          n      →      ∞            ∫          X            |        f          n            |      d  μ  =      ∫          X            |    f      |      d  μ.      |        f          n        −  f      |    ≤      |        f          n            |    +      |    f      |        ,  (  =      g          n        )(as in the theorem)and       lim          n      →      ∞            ∫                  R              (      |        f          n            |    +      |    f      |    )    d  λ  =  2      ∫                  R                  |    f      |      d  λ(So       ∫          X        g    d  μ  =      lim          n      →      ∞            ∫          X            g          n          d  μ)And each       f          n       and   f are integrable by assumption .So applying Dominated Convergence Theorem you get that      lim          n      →      ∞            ∫                  R                  |        f          n        −  f      |      d  λ  =  ∫      lim          n      →      ∞            |        f          n        −  f      |      d  λ  =      ∫                  R              0    d  λ  =  0

let f n </msub> : <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="doubl

Answered question

Answer & Explanation

New Questions in Pre-Algebra