Suppose $B$ is a ball in $\mathbb{R}^{n}$ ($ n\geq3 $), and $v_{t}$ a family of superharmonic functions on a neighborhood of the closure of $B$, and the index $t$ varies in an interval $(a,b)$ of $ \mathbb{R} $. We know that if $\mu_{t} $ is the Riesz measure associated with $v_{t}$, then we have \begin{align} \int_{B}\phi(x)d\mu_{t}(x)=a_{n}\int_{B}v_{t}(x)\Delta\phi(x)dx \end{align} for all $\phi\in C_{c}^{\infty}(B)$ ( $a_{n}$ is a positive constant). We know also that the potential given in the Riesz decomposition theorem is $$p_{t}(x)=\int_{B}\dfrac{d\mu_{t}(y)}{|x-y|^{n-2}}.$$
My question is: suppose $F(t)$ is the function on the right of the top =, and suppose I can prove that $t\mapsto F(t)$ is continuous on $(a,b)$ and bounded: $$ |f(x)|\leq M$$ ($M$ depends on $\phi$). Can I conclude that $t\mapsto p_{t}(x)$ is continuous for $x$ fixed?