I am reading Introduction to Model Spaces and their operators by Garcia, Mashreghi and Ross. In first page, the authors define what a support and carrier of a measure. The measures considered here are on the unit circle denoted by $\mathbb T$ and defined by $\mathbb T = \{ z\in \mathbb C : \lvert z \rvert =1 \}$. The Borel $\sigma$-algebra of $\mathbb T$ is the smallest $\sigma$-algebra containing the open arcs of $\mathbb T$.
Now, for a finite positive measure on $\mu$ on Borel $\sigma$-algebra, the support is defined to be the the complement of the union of all open subsets of $\mathbb T$ which have measure $0$. A Borel set $E \subset \mathbb T$ is called a carrier of $\mu$ if $\mu (E \cap A)=\mu (A)$ for each Borel subset $A$.
The authors claim thatthe support of $\mu$ is a carrier but I am unable to see how. Here's my attempt:
Let $\mathcal S$ denote the support of the measure $\mu$. Let $A$ be a Borel set of $\mathbb T$. We wish to show that $\mu (\mathcal S \cap A) = \mu (A)$. It is obvious that $\mu (\mathcal S \cap A) \le \mu (A)$. One needs to check that $\mu (\mathcal S \cap A) \ge \mu (A)$.
It is easy to see that $\mu (\mathcal S \cap A ) = \mu \left(\bigcap \left\{ A \cap F : F \text{ is a closed set satisfying } \mu(\mathbb T) =\mu (F) \right\}\right)$. I do not see how to proceed from here. Hints will be appreciated!
EDIT: Here's my attempt: After receiving a hint: I will show that the complement support is of measure $0$. Let $\mathcal U$ be the complement of the support. Since $\mathcal U$ is a open set in $\mathbb T$, $\mathcal U$ is countable union of open arcs $\mathcal C_i$ (one can mimic the same proof as that of for $\mathbb R$ see here). Since the open arcs contained in $\cal U$ must be of measure zero by definition of $\mathcal U$, we have that \begin{align*} \mu (\mathcal U) = \mu(\cup_{i=1}^{\infty} \mathcal C_i) \le \sum_{i=1}^{\infty} \mu(\mathcal C_i) = 0. \end{align*}
To finish the proof, notice that $\mu(A) = \mu(A \cap \mathcal S) + \mu (A \cap \mathcal U) \le \mu(A \cap \mathcal S) + \mu (\mathcal U)= \mu (A \cap \mathcal S)$.