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Could you please suggest a proof or a reference that shows a proof of the following equality (not "passing through the densities" as done in Definition of the total variation distance: $ V(P,Q) = \frac{1}{2} \int |p-q|d\nu$?), where left-hand side and right-hand side are both used as definitions of the total variation distance between two probability measures, $P$ and $Q$? \begin{equation} \max_{A \subseteq \mathcal{A}} \left| P(A)-Q(A) \right| = \frac{1}{2} \sum_{x \in \mathcal{A}} \left| P(x) - Q(x)\right| \end{equation}

Ommo
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    I think the proof is the same as this one which you have already seen. Just replace the densities $p, q$ with the PMFs $P$, $Q$, and replace integrals with sums. – angryavian Sep 18 '23 at 16:03
  • thanks angryavian! I have just found the proof by @am_rf24, from https://math.stackexchange.com/questions/3415641/total-variation-distance-l1-norm By chance, do you know any reference to a textbook or a paper? :-) – Ommo Sep 18 '23 at 16:28
  • I found another proof for my question (even though they use the supremum instead of the maximum): https://math.stackexchange.com/questions/3727561/total-variation-norm-of-probability-measures-related-to-l-1-norm – Ommo Sep 18 '23 at 21:39
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    It is actually pretty straightforward if you are not intimidated by the look. Just realize the maximum of the left hand side is achieved only when $P(x)-Q(x)$ has the same sign for all $x\in A$, otherwise the opposite signed number cancel each other and we can achieve a larger number by cutting out the $x$ with the opposite sign. Also keep in mind the $P(x)-Q(x)$ summed over all sample space is $0$. – Hans Sep 19 '23 at 21:14
  • Thanks @Hans, very kind! :-) – Ommo Sep 20 '23 at 07:35
  • In that proof, you should use densities.. I would like to see the proof without passing through densities (if possible) ...I think I would prefer this: https://math.stackexchange.com/questions/3727561/total-variation-norm-of-probability-measures-related-to-l-1-norm – Ommo Sep 20 '23 at 16:55

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I would read ``Markov Chains and Mixing Times'', second edition, by David A. Levin, Yuval Peres with contributions by Elizabeth L. Wilmer, Chapter 4. It is available online.

Did
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  • Thanks @Dimitrios D, Yes I have already read it :-) – Ommo Sep 20 '23 at 07:34
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    @Ommo is the question about more interesting/creative/what-adjective(?) proofs then, or perhaps proofs that lead to insights for a particular problem? – Did Sep 21 '23 at 00:02
  • Thanks for your comment Dimitros :-) I found very clear the proof & explanation provided by K. A. Buhr in "Total variation norm of probability measures related to 1 -norm?" (https://math.stackexchange.com/questions/3727561/total-variation-norm-of-probability-measures-related-to-l-1-norm). To reply to your question, i.e. "is the question about more interesting/creative/what-adjective(?) proofs ....", I do not think so :-) – Ommo Sep 21 '23 at 07:25