According to ProofWiki given a measure space $(\mathsf{X}, \mathcal{X}, \mu)$ and a subsigma algebra $\mathcal{Y}\subseteq\mathcal{X}$, the restriction of $\mu$ to $\mathcal{Y}$ is the measure $\mu\restriction_{\mathcal{Y}}:\mathcal{X}\to[0, +\infty]$ (important: it's domain is $\mathcal{X}$, not $\mathcal{Y}$!!) $$ \mu\restriction_{\mathcal{Y}}(\mathrm{A}) = \mu(\mathrm{A}) \qquad \forall \mathrm{A}\in\mathcal{Y} $$
QUESTION: The domain of $\mu\restriction_{\mathcal{Y}}$ is $\mathcal{X}$ but the definition only specifies how to measure sets $\mathrm{A}\in\mathcal{Y}$. How is the restriction defined for $\mathrm{B}\in\mathcal{X}$ with $\mathrm{B}\notin \mathcal{Y}$?
According to this post it does not assign measure zero to sets $\mathrm{B}$ above. Then what does it assign to them? It must assign something because it is defined over them.