Need help understanding definition of total variation metric of two probability measures

Fix a measurable space $$(X,\Sigma)$$. Let $$\mu:\Sigma\to\mathbb{R}$$ be some signed measure. Then the total variation norm is defined by $$\|\mu\|_{\textrm{TV}}=\mu_{+}(X)+\mu_{-}(X),\quad (*)$$ where $$\mu=\mu_{+}-\mu_{-}$$ is the Hahn decomposition of $$\mu$$. Now, let $$\mu,\nu:\Sigma\to[0,1]$$ be two probability measures. Then $$\|\mu-\nu\|_{\textrm{TV}}=2\max_{A\in\Sigma}(\mu(A)-\nu(A)),\quad (**)$$ To see this, recall that the Hahn decomposition $$\mu-\nu=(\mu-\nu)_{+}-(\mu-\nu)_{-}$$ has the form $$(\mu-\nu)_{\pm}(A)=(\mu-\nu)(A\cap A_{\pm}),\quad A\in\Sigma,$$ where $$A_{+}\cup A_{-}=X$$, and the choice of the sets $$A_{+}$$, $$A_{-}$$ is unique up to a set of zero measures $$|\mu-\nu|=(\mu-\nu)_{+}+(\mu-\nu)_{-}.$$ This apparently immediataly implies (**) with the maximum attained at the set $$A_{+}$$.

My question: I'm having a hard time connecting the definition of total variation norm in (*) with the arguments that lead to (**). Any help is greatly appreciated.

• what is the reference you are using here?
– Ommo
Commented Sep 14, 2023 at 8:41

From the Hahn decomposition theorem, you can check that the definition

$$\|\mu\|_{\textrm{TV}}=\mu_{+}(X)+\mu_{-}(X),\quad (*)$$

can be rewritten as

$$\|\mu\|_{\textrm{TV}}=\max_{A \in \Sigma}\mu(A)-\min_{A \in \Sigma}\mu(A)$$

with the max and min achieved at $$A_+$$ and $$A_-$$ respectively.

So for the difference of two probability measures,

\begin{align*} \|\mu-\nu\|_{\textrm{TV}}&=\max_{A \in \Sigma}(\mu(A)-\nu(A))-\min_{A \in \Sigma}(\mu(A)-\nu(A))\\ &=(\mu(A_+) - \nu(A_+)) - (\mu(A_-)-\nu(A_-))\\ &=(\mu(A_+) - \nu(A_+)) - ((1-\mu(A_+))-(1-\nu(A_+)))\\ &=2(\mu(A_+) - \nu(A_+))\\ &=2\max_{A \in \Sigma}(\mu(A)-\nu(A)) \quad (**) \end{align*}

• any reference for this proof?
– Ommo
Commented Sep 18, 2023 at 10:12
• No, sorry; I'd assume it's in a textbook somewhere, but I don't have a citation. @limone