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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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In the previous section, we defined the Lebesgue integral and the expectation of random variables and showed basic properties. However the additive property of integrals is yet to be proved. In addition, since our major interest throughout the textbook is convergence of random variables and its rate, we need our toolbox for it.
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In this section, we define the expectation of a random variable as the Lebesgue integral with respect to the probability measure. First, I will introduce the standard machine in measure theory and use it to define the Lebesgue integral. Next, the definition of expectation will be discussed in terms of the Lebesgue integral.
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We take a closer look at random variables and random elements.
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