MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Let f : R &#x2192; R be a continuous bounded function, ( X k ) k a sequence of i.i.d random variables such that

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MathJax(?): Can&#039;t find handler for document MathJax(?): Can&#039;t find handler for document Let   f  :      R    &amp;#x2192;      R   be a continuous bounded function,   (      X    k        )    k   a sequence of i.i.d random variables such that

Blaze Frank · Accepted Answer

You have already found that   f  (  x  +  y  )  =  f  (  x  ) for       P                  X        1            -a.e.   y  ∈      R  . Now, let   y  ∈      s    u    p    p        P                  X        1            . Then       P                  X        1              (  y  −  1      /    n  ,  y  +  1      /    n  )  &amp;gt;  0 for every   n  ∈      N  . Therefore, there must exist       y    n    ∈  (  y  −  1      /    n  ,  y  +  1      /    n  ) such that   f  (  x  +      y    n    )  =  f  (  x  ). Obviously       y    n    →  y and since   f is continuous we have&amp;lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; &quot;&amp;gt;  f  (  x  +  y  )  =      lim          n      →      ∞        f  (  x  +      y    n    )  =  f  (  x  )  .

Let f : <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R </mrow>

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