I asked this at math.stackexchange, but nobody answered.

Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ${\mathcal C}(K)^*$ the Banach dual space of measures.

For each measure $\mu\in{\mathcal C}(K)^*$, $\mu\ge 0$, consider the natural mapping $$ \varPhi_\mu: L_1(\mu)\to {\mathcal C}(K)^*\quad\Big|\quad \varPhi_\mu(f)=f\cdot\mu,\quad f\in L_1(\mu), $$ or, in other words, $$ \varPhi_\mu(f)(x)=\int_K x(t)\cdot f(t)\cdot\mu(d t),\quad f\in L_1(\mu),\quad x\in {\mathcal C}(K). $$ Let $p:{\mathcal C}(K)^*\to{\mathbb C}$ be a linear functional, which is continuous on each subspace $L_1(\mu)$, i.e. for any $\mu$ the composition $p\circ\varPhi_\mu$ is continuous (=bounded) on the Banach space $L_1(\mu)$ (with the usual integral norm).

Is $p$ continuous on ${\mathcal C}(K)^*$? (Equivalently, is $p$ an element of ${\mathcal C}(K)^{**}$?)