In this section I would like to cover the Borel-Cantelli lemmas, or B-C lemmas for short. Borel-Cantelli lemmas are quintessential tools for analysis of tail events and deriving almost sure convergence from $P$-convergence.
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We say a random variable $X_n$ converges in probability (or $P$-converges) to another random variable $X$ and write $X_n \overset{P}{\to} X$ if $\lim_n P(|X_n - X| > \epsilon) = 0$ for all $\epsilon >0.$ We can also define convergence in probability to a constant by letting $X = c\in\mathbb{R}$. It is easy yet useful to know that $X_n \overset{L^p}{\to} X$ with $p > 0$ implies $X_n \overset{P}{\to} X$ by Markov-Chebyshev inequality with $\varphi(x) = |x|^p$.¹
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First chapter was about the essential of measure theory. We especially focused on important result for finite measure or at least $\sigma$-finite measures. We defined a probability space as a measure space and a random variable as a measurable function in it.
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Let $X, Y$ be random variables on $(\Omega_1, \mathcal{F}_1, P_1)$ and $(\Omega_2, \mathcal{F}_2, P_2)$ respectively. In order to well-define sets such as $\{ X + Y \le 0 \}$, we should consider a random vector $(X,Y)$ on a product space $\mathbb{R} \times \mathbb{R}$ since $+: \mathbb{R}\times\mathbb{R} \to \mathbb{R}$ is defined this way. In addition, to measure the probability of such sets, we also need to define another product probability space $(\Omega, \mathcal{F}, P)$ where $\Omega = \Omega_1 \times \Omega_2$. Main question to answer in this subsection is: how can we define a proper $\sigma$-field $\mathcal{F}$ and a product measure $P$ on such $\Omega$?
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In the previous section, we defined Lebesgue integrability and write $f \in L^1$ for such function $f$. We also defined the $L^p$-norm. $L$ in these notations stands for “Lebesgue” and $L^p$ for $p \ge 1$ becomes a space of functions that is integrable in the order of $p$. We name it $L^p$ space.
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