### 1.7. Product space

probability Durrett

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 $\Omega$?

### 1.6. $L^p$ space

probability Durrett

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.

### 1.5. Convergence theorems and elementary inequalities

probability Durrett

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.