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Just so that everyone knows what we are talking about here, let me rephrase in more familiar notation. Suppose $(\Omega, \mathcal{F}, P)$ is a probability space, and $(M, \mathcal{M})$ is a measurable space. If $X : \Omega \to M$ is a random variable (i.e. a $(\mathcal{F}, \mathcal{M})$-measurable function), it induces a pushforward measure on $(M, ...


Possibly you're thinking of the Poisson distribution. Suppose you have a one-in-$1{,}000{,}000$ chance of success on each trial, and there are $3{,}600{,}000$ trials. The expected number of successes is then $3.6$. If we ask for the probability that there are exactly $5$ successes, we get $$ \frac{3.6^5 e^{-3.6}}{5!} = \frac{3.6^5 e^{-3.6}}{120} \approx ...


It seems like you may be referring to the Law of Large Numbers. The Wikipedia page linked to gives a good explanation, from what I can see. See also: the relevant part of this SE answer for more information on the theorem, and why it is important, this SE question for another good, layman oriented explanation of the theorem, the top-rated answer to this SE ...


The correct approach is always to consider the common sample space. If the events given are independent then the common sample space can be constructed. If they are not independent then either the dependence structure is given on a common sample space or the common sample space is not defined. In the case of independent dice rolling the sample space is ...


Given a measureable $f:(X,\mathcal{M},\mu)\to(Y,\mathcal{N})$ (a measure space to a measurable space) one can define a measure $\nu=f_{*}\mu$ on $(Y,\mathcal{N})$ by $$ \nu(B)=\mu(f^{-1}(B)). $$ Both $f$ and $\mu$ are necessary to define it, so $f_{*}\mu$ seems like good notation to me.


$\phi$ holds almost surely on $S$ means that there is a null set $N$ such that $$\omega \in S \setminus N\Rightarrow \phi(\omega) \text{ holds} $$ This is your Option $3$. Note that this notion is only interesting when $P(S)>0$. That is, on a null set, every property holds almost surely. The idea to consider regular conditional probabilities is not ...


This is the Borel sigma algebra on the Baire space which is homeomorphic to irrationals. So there can be at most continuum many such sets. Here's the wiki article about Baire space: https://en.m.wikipedia.org/wiki/Baire_space_(set_theory)

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