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mona123

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decimal expansion

hi
i find this problem difficult to solve can someone help me
Let x ∈ [0, 1] have the decimal expansion x = .x1x2x3 . . . . Define fn on [0, 1] by
fn(x) = xn. Prove that fn is measurable. (If x has two decimal expansions, use the non-terminating expansion for defining fn.)
thanks

Fred

Administrateur

Inscription : 26-09-2005

Messages : 7 352

Re : decimal expansion

Since [tex]f_n[/tex] has values in [tex]\{0,\dots,9\}[/tex], it suffices to show that [tex]f_n^{-1}(\{k\})[/tex] is measurable for any [tex]k\in\{0,\dots,9\}[/tex]. This can be done by observing that [tex]f_n^{-1}(\{k\})[/tex] is open for any n and any k. To fix the notations, let us assume that n=2 and k=1. Then
[tex]f_n^{-1}(\{1\})=[0,01;0,02[\cup [0,11;0,12[\cup \dots\cup [0,91;0,92[ [/tex]

Fred.