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I see a lot of examples of limit theorems in terms of functions, and sequences of functions. But I think the transition from the general measure space to the probability space ($([0,1],\mathcal{B}[0,1],\text{Leb})$, say) and random variables is confusing.

Taking a sequence of random variables $X_n(\omega)=n^2\mathbb{1}_{(0,n^{-1})}$, I am now presented with the variable $\omega$ - and what isn't clear is how we are able to use analysis in this case. Taking the supremum, for example,

$\sup_n X_n= \sup_n\left( n^2 \mathbb{1}_{(0,n^{-1})}\right)$.

So it would seem that if $\omega$ is in the set then the value is equal to $n$, but the interval gets really small, so saying that the supremum is $\infty$ is what I am unsure about because in taking $n\to\infty$ ensures that this set is the empty set, $\emptyset$. Also, taking the supremum over $n$ puts no condition on $\omega$ being in the set so how is there a supremum when $X_n$ can take two possible values for any $n$ which is not determined by me beforehand?

With the convention of taking $0\cdot \infty=0$. Does this mean that $\lim_{n\to\infty} X_n=\infty \cdot \mathbb{1}_{\emptyset}=\infty\cdot0=0$ is the correct way to show that $X_n\to0 \text{ a.s.}$? This feels too easy because I know that I need to show that for every $\omega \in\Omega$, $ \lim_{n\to\infty}\mathbb{P}(|X_n|>\epsilon)=0$. This leaves me the task of justifying $\lim_{n\to\infty}\mathbb{P}(\mathbb{1}_{(0,n^{-1})}>\frac{\epsilon}{n^2})=0$, is it correct that this is simply $\lim_{n\to\infty}\mathbb{P}((0,n^{-1}))=0?$

So my question is, how does the analysis used in measure theory correspond to the case of random variables? Measures, probability measures, expectations and integrals all feel like they are being used interchangeably I need to distinguish between them and know which to use in certain cases.

I think I need some examples also, the notation in one text will differ considerably from another and a method used in one will be to do with a general measure space and I cannot transfer to probability. I have found a lot of lecture notes online but few examples.

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2 Answers 2

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Don't let the terminology and notation used in probability theory confuse you. A probability space is just a measure space equipped with a measure that gives the whole space measure $1$; a random variable is just a measurable function where the domain is a probability space and expectation is just an integral where we integrate with respect to a probability measure. And it's very common to denote the probability space with $\Omega$ and its members $\omega$ - often even when we choose $\Omega$ to be something otherwise very familiar, like $[0,1]$. The reasons for such different conventions compared to measure theory and Analysis are partly historical inertia and partly to employ extremely useful intuitive notions - everyone has some idea what 'probability', 'event' and 'expectation' mean even if they don't know any measure and probability theory.

Everything you know about measure theory holds also in probability theory and often in a stronger form since probability spaces are finite measure spaces. Dominated convergence, Fubini and $L^p$ spaces are used all the time in probability theory. In addition there are concepts specific to probability that one doesn't usually see elsewhere in measure theory and Analysis: independence, filtration, martingale... Many would probably say that it is these objects of study specific to probability theory that defines it; not the terminology and notation.

To ease your discomfort with '$\omega$', '$\Omega$', 'random variables' and so on take any example and just change them to '$x$', '$X$', 'measurable function' etc. Then work through the example and when you understand it just change them all back and look at the example again. You will get used to them soon enough.

All the best to you in your studies of probability theory!

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You say $X_n = n^2 1_{(0,n^{-1})}$. I assume the measure space here is $[0,1]$ with Lebesgue measure. For any particular $\omega \in (0,1)$, $X_n(\omega) = n^2$ if $0 < \omega < n^{-1}$, i.e. $n < 1/\omega$, but $X_n(\omega) = 0$ if $n \ge 1/\omega$. So the sequence $X_n(\omega)$ is $0$ from some point on, and $\lim_{n \to \infty} X_n(\omega) = 0$. You don't have to do anything about $P(|X_n|>\epsilon)$ unless you're investigating convergence in probability.

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The connection there between $\omega$ and $n$ I would not have anticipated. The $\omega$ always makes me feel uneasy. I thought that you needed to evaluate probabilities in almost sure convergence –  shilov Mar 11 '13 at 20:51
Almost sure convergence is pointwise convergence on a set that has probability $1$. So in general you don't have to evaluate any probabilities except perhaps $0$ or $1$. –  Robert Israel Mar 12 '13 at 3:13
Thank you for the help! –  shilov Mar 14 '13 at 2:17

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