測度

\[ \begin{aligned} &X : \text{set} \\ &\boldsymbol{B} \subset \mathfrak{P}(X) : \text{completely additive class of sets of } X \\ &\mu : \boldsymbol{B} \rightarrow \mathbb{R} \end{aligned} \]

\[ \begin{aligned} &\text{(1)} \quad &&^{\forall} A \in \boldsymbol{B} , \quad 0 \leqq \mu(A) \leqq \infty \\ &\text{(2)} \quad &&\mu(\phi) = 0 \\ &\text{(3)} \quad &&^{\forall} n \in \mathbb{N}, \quad A_{n} \in \boldsymbol{B} \\ &&&i \neq j, \quad A_{i} \cap A_{j} = \phi \Longrightarrow \mu \left( \overset{\infty}{\underset{n=1}{\bigcup}} A_{n} \right) = \sum_{n=1}^{\infty}{ \mu(A_{n}) } \end{aligned} \]

\[ \begin{aligned} &\mathbb{R}^{N} : n \text{-dim Euclidian space} \\ &\mathcal{B}^{(N)} : \text{Borel sigma-algebra of } n \text{-dim Euclidian space} \\ &( \mathbb{R}^{N} , \mathcal{B}^{(N)} ) : \text{mesurable space} \\ &\mu \left( \prod_{i=1}^{N} {( a_{i}, b_{i} ]} \right) = \prod_{i=1}^{N} {(b_{i} - a_{i})} \end{aligned} \]

\[ \begin{aligned} &( \Omega, \mathcal{F} ) : \text{mesurable space} \\ &^{\forall}{A} \in \mathcal{F}, \quad 0 \leqq \mu (A) \leqq 1 \\ &\mu (\Omega) = 1 \\ \end{aligned} \]