Why power set of $Omega$ (set of all outcomes) can't always be used as field in probability space ($Omega,mathcal{F}, P$)?

I guess it’s because, for example, when $Omega=mathbb{R}$ (denote its size as $|mathbb{R}|$), its power set would be too large, with its size equals $2^{|mathbb{R}|}$, and though we can find a nowhere zero function $f$ of domain as large as $mathbb{R}^n$, ${{0,1}}^{mathbb{R}}$ is too much larger than $mathbb{R}^n$ that we can’t find $f$ such that $int_{2^{mathbb{R}}} f= 1$.

Because probability for a continuous variable is $0$ at all points and this fact would make things confusing, let me make change the discussion to one on a discrete variable.

By dissecting ${mathbb{R}}$ into countable ($|mathbb{N}|$) closed intervals, we dissect ${{0,1}}^{mathbb{R}}$ into a collection $X$ of $2^{|mathbb{N}|}$ closed intervals, We can regard the integral as ‘infinite’ ($O(2^{N})$, larger than the size of a typical inf series ) series $sum_{qin {{0,1}}^{mathbb{N}}} a_q$, where $a_q$ is probability of $q$th closed intervals $delta omega_q$ in $X$, and $a_q= int_{delta omega_q}f$. So the hypothesis in the last paragraph can be rephrased as we can’t find we can’t find a function $a(q)$ of $q$ such that $sum_{{{0,1}}^{mathbb{N}}} a(q) = 1$.

Well, I myself think this reasoning is flawed. Perhaps we can define ${a_q}$ to be a sequence that converges so fast that the inf series $sum a_q$ converges even if there are uncountable items to be summed. Is that right?

Or perhaps the reason is not as above, but that if we let the power set be set $mathcal{F}$ of events , it would cause contradictions in calculating probability of intersections/unions of events?