Does the the equivalence of Total variation distance formulas presented here([https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf](https://ece.iisc.ac.in/%7Eparimal/2019/statphy/lecture-14.pdf)) assumes that the two distributions are symmetrical ?



Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?
All the expressions for the total variation distance given in Definition 1.1 and Propositions 1.2, 1.4, 1.7 in the linked lecture notes hold in general, without any symmetry assumptions.


Indeed, the proofs there do not use any symmetry assumptions.


Moreover, all these expressions hold when $\mathscr X^N$ is replaced by (say) any countable set (say $\mathscr Y$) without any structure, so that no symmetry condition can be possibly defined on $\mathscr Y$ in general.


