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During my research, someone mentioned the following metric on the space of real sequences: $$d(x,y)=\sum_i 2^{-i}\frac{|x_i-y_i|}{1+|x_i-y_i|}$$ I was wondering whether this metric had a name or there was a source mentioning this metric?

The commenter also seemed to imply that with this metric, you can 'extend' the lebesgue measure on binary sequences to this space to a probability measure on the space of real sequences. However I don't see how these ideas are linked, as I don't even see a 'natural' probability measure on $\mathbb{R}$ associated with this metric (if that even makes sense).

Anton V.
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    It may be called the product metric because (with the usual topology on $\Bbb R$) it generates the (Tychonoff) product topology on $\Bbb R^{\Bbb N}$ which is sometimes called the topology of pointwise convergence: Consider real sequences as functions from $\Bbb N$ to $\Bbb R.$ Then $\lim_{n\to\infty}d(f_n,f)=0$ iff $\forall k\in\Bbb N,(\lim_{n\to\infty}f_n(k)=f(k),).$ – DanielWainfleet Jan 18 '22 at 15:03

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This is just a transformation of the standard metric on $\mathbb{R}$. If $$d(x,y)$$ is a metric, then $$ \dfrac{d(x,y)}{1 + d(x,y)}$$ is also a metric. Here $d(x,y)=|x-y|$.

The scaling by powers of 2 ensures absolute convergence. See, for example here and here

AlDante
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