Restricting a σ-Algebra or a Measure Twice

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, for classes $\mathcal{F}$, $\mu$ and $A$ let

$$\mathcal{F}_{A}=\{C\in\mathcal{F}:C\subseteq A\},\qquad \mu_{A}=\mu|_{\mathcal{F}_{A}},$$

formed by restricted class abstraction, with $\subseteq$ the subclass relation and $\mu|_{\mathcal{F}_{A}}$ the restriction of $\mu$ to $\mathcal{F}_{A}$.

(i) For all classes $\mathcal{F}$, $X$ and $A$ with $A\subseteq X$:

$$(\mathcal{F}_{X})_{A}=\mathcal{F}_{A}.$$

(ii) For all classes $\mathcal{F}$, $\mu$, $A$ and $B$ with $B\subseteq A$, the restriction of $\mu_{A}$ to $(\mathcal{F}_{A})_{B}$ is $\mu_{B}$:

$$(\mu_{A})|_{(\mathcal{F}_{A})_{B}}=\mu_{B}.$$