Correct notation for restriction of measure

Question

Suppose $(\Omega,\Sigma,\mu)$ is a measure space and $F\subset \Omega$, $F\in\Sigma$. Then we can define a new measure $\mu_{F}$, the restriction of $\mu$ to $F$. So this measure is only defined on the trace sigma algebra $\Sigma_F$ or is it defined on $\Sigma$, i.e. what is the correct definition, $\mu_F:\Sigma_F\to[0,\infty]$ or $\mu:\Sigma\to[0,\infty]$, but $\mu(A)=0,\quad\forall A\in \Sigma\setminus\Sigma_F$? And so $\mu_F=\mu, \quad\forall A\subset F, A\in \Sigma$, too?

Answer 1

$\Sigma_F =\{A\subset F: A \in \Sigma\}\equiv \{A\cap B: B \in \Sigma\}$ and $\mu_F$ is defined on $\Sigma_F$ by $\mu_F(A)=\mu(A)$ whenever $A \in \Sigma_F$. This makes $(F,\Sigma_F,\mu_F)$ a new measure space.