> [!definitionb] Definition > > Let $\Omega$ be a [[Set|set]], $\Sigma \subseteq \Omega$ be a subset where a notion of measure[^1] $I: \Sigma \to [0, \infty]$ is defined. > > Let $x \in \Omega$ and suppose that there are measurable collections $\ol{\Sigma}_x, \ul{\Sigma}_x \subseteq \Sigma$ that approximate $x$ from below and above, respectively. Then the [[Supremum and Infimum|supremum and infimum]] of the lower and upper collections > $ > \ol{I}(x) = \sup_{\phi \in \ol{\Sigma}_x}I(\phi) \le \inf_{\phi \in \ul{\Sigma}_x}I(\phi) = \ul{I}(x) > $ > are the **lower** and **upper** approximations of $I(x)$. Generally, If > $ > \ol{I}(x) = I(x) \text{ or } \ul{I}(x) = I(x) \quad \forall x \in \Sigma > $ > the approximations agree with $I$, then $\ol{I}$ and/or $\ul{I}$ **extend** $I$. > > The idea of lower and upper approximations is commonly used in analysis and measure theory. In particular, the [[Riemann Integral]] (Darboux's characterisation), the [[Outer Measure|outer measure]], [[Carathéodory's Theorem]], [[Carathéodory's Extension Theorem]], [[Integral]] for non-[[Simple Function|simple]] functions, the [[Monotone Convergence Theorem|monotone convergence theorem]] and [[Fatou's Lemma]] all make use of this concept. > [!theorem] > > Let $x, y \in \Omega$, then $\ol{\Sigma}_x \subseteq \ol{\Sigma}_y \Rightarrow \ol{I}(x) \le \ol{I}(y)$, and $\ul{\Sigma}_x \subseteq \ul{\Sigma}_y \Rightarrow \ul{I}(x) \ge \ul{I}(y)$. > [!theorem] > > Let $x \in \Omega$ and $c \in [0, \infty]$. If $I(\phi) \le c \forall \phi \in \ol{\Sigma}_x$, then $\ol{I}(x) \le c$. If $I(\phi) \ge c \forall \phi \in \ul{\Sigma}_x$, then $\ul{I} \ge c$. > [!theorem] Controlling the Lower Limit > > Let $x \in \Omega$, $\seq{x_n}$, and $\seq{\phi_n} \subseteq \ol{\Sigma}_x$ be a supporting [[Sequence|sequence]] such that $I(\phi_n) \upto \ol{I}(x)$. If for any $n \in \nat$, $\exists \psi_n \in \Sigma_{x_n}$ such that $I(\phi_n) \le I(\psi_n)$ eventually, then > $ > \ol{I}(x) = \limv{n}I(\phi_n) \le \liminf_{n \to \infty} I(\psi_n) \le \liminf \ol{I}(x_n) > $ > *Proof*. Since $I(\phi_n) \le I(\psi_n) \le I(x_n)$ eventually, we have > $ > \liminf_{n \to \infty}I(\phi_n) \le \liminf_{n \to \infty} I(\psi_n) \le \liminf_{n \to \infty} \ol{I}(x_n) > $ > by the [[Order Limit Theorem|order limit theorem]]. Since $I(\psi_n) \upto \ol{I}(x)$, > $ > \ol{I}(x) = \limv{n}I(\phi_n) \le \liminf_{n \to \infty} I(\psi_n) \le \liminf \ol{I}(x_n) > $ [^1]: I don't have a better word than measure. This isn't really a measure, but we'll work with it.