Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric on the latter space $$ d(\mu,\nu) :=\sup\left\{\int_\Omega f\;\mathrm d\mu - \int_\Omega f\;\mathrm d\nu:f\in \mathcal B_1(\Omega)\right\} $$ where $\mathcal B_1(\Omega)$ is the space of all Borel functions on $\Omega$ whose absolute value does not exceed $1$.
Clearly, any linear functional $\phi:\mathcal P(\Omega)\to \Bbb R$ of the form $$ \phi(\mu) = \int_\Omega(c+f)\mathrm d\mu \tag{1} $$ for any $f\in \mathcal B_1(\Omega)$ and $c\in \Bbb R$ satisfies the following Lipschitz condition $$ |\phi(\mu) - \phi(\nu)| \leq d(\mu,\nu). \tag{2} $$ Is that true that any linear functional $\phi$ that satisfies $(2)$ can be represented as $(1)$?
Borel space is a topological space homeomorphic to a Borel subset of a complete separable metric space.
The answer below does not seem to be correct according to its author.
Let $\mu^d$ be the restriction of a measure $\mu\in\mathcal{P}(\Omega)$ to its atoms ($\mu^d(A)$ is the maximum of $\mu(A\cap S)$ over countable $S\subseteq\Omega$). Then a counterexample is given by $$ \phi(\mu)=\mu^d(\Omega)=\max\left\{\mu(S)\colon S\subseteq\Omega\textrm{ is countable}\right\}. $$