$\sigma$ algebra and complete lattice If Ω is not finite, let’s say it is countable, then is any σ-algebra defined on it a complete lattice?
Here is what I have been thinking suppose $\Omega$ is finite then the set has Greatest Element and the least element if I am able to prove that any subset has a lower bound then it would be a complete lattice. But I am not able to figure out how to do that and also when the $\Omega$ is not finite.
 A: $\hspace{0.44cm}$Below is my attempt. Although it is too late to answer your question but it would be helpful if any mistakes are spotted.
$\hspace{0.44cm}$Let $S$ be the given infinite set and $\Sigma$ be the given $\sigma$-algebra. Assume there is a finite positive real-valued set function $\mu$ defined on $\Sigma$ such that $(S, \Sigma, \mu)$ is a finite measure space. Let $\{A_{\lambda}\} \subset \Sigma$. Since $\{\mu(A_{\lambda})\} = \{\|\chi_{A_{\lambda}}\|_1\}$, we then want to show the function $\chi_A$ where $A = \bigcup A_{\lambda}$ is $\mu$-measurable and hence $A \in \Sigma$. Then we can find a sequence $\{A_n\}\subseteq\{A_{\lambda}\}$ such that $\mu(A_n) \uparrow \mu(A) = \sup_{A_{\lambda}}\mu(A_{\lambda})\,\implies\,\|\chi_{A_n} - \chi_A\|_1 \rightarrow 0$ (since $\mu$ positive) $\implies\,\chi_{A_n}$ converge to $\chi_A$ in $\mu$-measure (definition can be found here).
$\hspace{0.44cm}$One can refer to this paper for definitions of totally measurable functions and measurable functions. Briefly speaking in a positive finite measure space, a sequence of simple functions can only converge in measure to a totally-measurable function. If that limit function is also a characteristic function, then the set where it obtain $1$ will be measurable. Hence here we can conclude $A \in \Sigma$.
$\hspace{0.44cm}$Pick $B \in \Sigma$ and fix $b \in B$. Then the Dirac delta measure $\delta_b$ is a finite positive real-valued measure (the idea here comes from this post) and the existence of $\sup\{A_{\lambda}\}$ in $\Sigma$ follows the remark above and so does the existence of $\inf\{A_{\lambda}\} = S \backslash\sup\{A_{\lambda}^c\}$.
$\hspace{0.44cm}\Large\spadesuit$Now I am also curious if $\Sigma$ admits any other complex measures $\lambda$ (i.e. $\Sigma$ is the set of all $\lambda$-measurable sets). If so, then $\arctan \vert\lambda\vert$ is also a positive real-valued measure.
