What are boundary effects in measure-theory? I often read the term "boundary effects" which seem to be the reason to look at the interior $A^°$ and closure $\bar{A}$ of Borel subsets $A$ separately. What is so special about the boundary and what cases are problematic?
Edit: I suppose this is related to the properties of weak convergence of the probability measures $\{\mu_n\}$ to $\mu$ on $S$:
\begin{align*}
\lim\sup_n\mu_n(F) &\leq \mu(f), &\quad &\text{for all closed } F\subseteq S\\
\lim\inf_n\mu_n(G) &\geq \mu(G), &\quad &\text{for all open } G\subseteq S\\
\end{align*}
 A: The following arguments are taken from Convergence of Probability Measures by Patrick Billingsley. The Portemanteau Theorem says (among other things) that a sequence $P_n$ of probability measures on a metric space $(S,\mathcal{S})$ (equipped with its Borel $\sigma$ algebra) converges weakly to a probability measure $P$ on that space if and only if $P_n(A)\to P(A)$ for all $A\in\mathcal{S}$ whose boundary $\partial A$ satisfies $P(\partial A)=0$. It also says that each of your two conditions could be taken as the definition of weak convergence.
Suppose that $A\in\mathcal{S}$. Let us take the two conditions you mention (they are equivalent) as definition of weak convergence and show that they imply that $P_n(A)\to P(A)$ for all $A$ with $P(\partial A)=0$ if $P_n\Rightarrow P$.
Then $P(\bar{A})\geq \limsup_n P_n(\bar{A})$ by the first condition since $\bar{A}$ is closed. Now, $\limsup_n P_n(\bar{A})\geq \limsup_n P_n({A})$ because $\bar{A}\supset A$. Of course $\limsup_n P_n({A})\geq \liminf_n P_n({A})$. Because $A\supset A^{\circ}$, we get $\liminf_n P_n({A}) \geq \liminf_n P_n({A^{\circ}})$, and by the second condition we see that $\liminf_n P_n({A^{\circ}})\geq P(A^{\circ})$. 
If $A$ is a set whose boundary has no $P$-mass, then $P(\bar{A})=P(A)=P(A^{\circ})$, and $P_n(A)$ indeed converges to $P(A)$.
