In the theory of knowledge and common knowledge, an event is any subset $E$ of a sample space $\Omega$. For a given partition $\mathcal{P}$ of $\Omega$, it is said that an agent knows event $E$ at $\omega\in\Omega$ if $\mathcal{P}(\omega)\subseteq E$, where $\mathcal{P}(\omega)$ denotes the partition cell to which $\omega$ belongs.
In measure theory, an event is an element of a $\sigma$-algebra. Can we say that knowledge of an event is equivalent to saying that it belongs to the $\sigma$-algebra generated by the given partition $\mathcal{P}$ or are events in measure theory and in common knowledge theory different objects?
I've figured it out. An event is any element from a given $\sigma$-algebra, not the one generated by the partition. It is not true that agent knows $E$ whenever $E$ belongs to the $\sigma$-algebra generated by the partition. Here is a simple example.
Let $\Omega=\{\omega_1,\omega_2,\omega_3,\omega_4\}$. Endow $\Omega$ with $\sigma$-algebra $\mathcal{F}=2^{\Omega}$. Consider the partition $\mathcal{P}=\{\{\omega_1, \omega_2\}\{\omega_3, \omega_4\}\}$. It generates the following $\sigma$-algebra: $\sigma(\mathcal{P})=\{\emptyset, \Omega, \{\omega_1, \omega_2\}\{\omega_3, \omega_4\}\}$. Now consider event $E=\{\omega_1, \omega_2, \omega_3\}$. At $\omega=\omega_1$, agent knows $E$ because $\mathcal{P}(\omega_1)\subset E$. But it is not true that $E\in\sigma(\mathcal{P})$.