Given a measurable space $(X,M)$ let $\delta_a$ and $\delta_b$ be the pointmass measures at points $a$ and $b$ in $X$ (suppose $X$ has at least two points to avoid trivialities). It easy to see that $\|\delta_a+\delta_b\|=2$. Out of curiosity, I'm trying to find an example of a measurable space where $\|\delta_a-\delta_b\|\not=2$.
2026-04-03 06:06:09.1775196369
Looking for an example of a measurable space where $\|\delta_a-\delta_b\|\not=2$ for pointmass measures
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What you need is a $\sigma$-algebra where $a$ and $b$ do not live in distinct elements of a partition. For instance let $X=\{a,b,c\}$, and $$ M=\{\varnothing, X, \{a,b\}, \{c\}\,\}. $$ This is a $\sigma$-algebra (contains complements, unions, intersections). And, for any $Y\in M$, you have $\delta_a(Y)=\delta_b(Y)$. So $\delta_a-\delta_b=0$.