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1 1 $mathcal F$ is a set of events ( $sigma$ -algebra). Can you give an example for some algebra $mathcal A$ over $mathbb N$ a non-zero finite additive measure $mu $ on this algebra, which has a countably additive extension to the $sigma$ -algebra generated by this algebra, moreover, when shifting any set $A ∈ mathcal F$ by an integer $n$ , for the so obtained set $A + n$ was fulfilled: $A + n ∈ A$ , $mu (A + n) = mu $ (A)? probability-theory measure-theory examples-counterexamples outer-measure share | cite | improve this question edited Dec 3 at 9:22

1 It is well known that $$frac{x-pi}{2}=-sum_{kgeq 1}frac{sin{kx}}{k}forall xin(0,tau),$$ which gives $$frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}=sum_{kgeq 1}frac{cos(kx)}{k^2}.$$ Note that $$textrm{Li}_2(e^{ix})=sum_{kgeq 1}frac{cos(kx)+isin(kx)}{k^2}$$ This means that $$mathfrak{R}textrm{Li}_2(e^{ix})=frac{x^2}{4}-frac{pi x}{2}+frac{pi^2}{6}$$ unless I'm wrong on one of the above statements. Now, with the previous restrictions on $x$ , we have that this formula is valid for all complex numbers lying on the complex unit circle as inputs. My question is: is there a way to define something along these lines (a finite degree polynomial, preferably) for complex inputs of the dilogarithm that don't necessarily lie on the complex unit circle? More specifically, $$mathfrak{R}textrm{Li}_2(re^{ix})=?$$