Measure theory problems. Prove or disprove the following: a)If $\mathscr A$ is a $σ$-Algebra on $Ω$ then {$Ω$ \ $A$ : $A$ element of $\mathscr A$} too. b)A $σ$-Algebra with 3 elements exists. c)A measure $μ$ on $P(\mathbb R)$ with $μ$({x}}=$1$ exists. * * * a) Isn't this equal to Ω \ $\mathscr A$? If so then this is the second property of a $σ$-Algebra. b) Ω={1,2,3}, $\mathscr A$={{1},{2},{3}} is right? c) I don't know I just know that for a Lebesgue-Measure $μ$({x}} is equal to $0$. Don't know if any measure exists that can satify c).

a) A $\sigma$-algebra is stable under complementation, hence $\\{\Omega\setminus A,A\in\mathcal A\\}=\mathcal A$ (it cannot be equal to $\Omega\setminus \mathcal A$, because $\mathcal A$ is a _class of subsets of $\Omega$_ , while $\Omega$ is a set.

b) Your $\mathcal A$ is not a $\sigma$-algebra (it should at least contain $\Omega$). If $\mathcal A$ is a $\sigma$-algebra with more than two elements, there is $A\in\mathcal A$ which is neither the whole set nor empty. But this is also the case for $\Omega \setminus A$.

c) Consider the counting measure (which assigns to each sets its number of elements if the set if finite, and $+\infty$ otherwise).