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[itex]T : X \rightarrow X[/itex] is a measure preserving ergodic transformation of a probability measure space [itex]X[/itex]. Let [itex]V_n = \{ g | g \circ T^n = g \} [/itex] and [itex]E = span [ \{g | g \circ T = \lambda g, [/itex] for some [itex]\lambda \} ][/itex] be the span of the eigenfunctions of the induced operator [itex]T : L^2 \rightarrow L^2[/itex], [itex]Tf = f \circ T[/itex].

Problem:

I was reading this paper by Fursternberg and Weiss where they implicitly claim if [itex]f \perp E[/itex] then [itex] f \perp V_n[/itex]. However, I don't see how this isso.

Some help would be greatly appreciated. : )

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# Projection to Invariant Functions:

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