$\textbf{Measure}$
Let $X$ be a set and $\mathscr{A}$ a $\sigma$-Algebra over $X$. A function $\mu:\mathscr{A}\to[0,\infty]$ is called a measure if it satisfies the following properties:
1) Null empty set: $\mu(\varnothing)=0$
2) Countable additivity (or $\sigma$-additivity): For any infinite sequence $\{A_i\}$ of pairwise disjoint sets in $\mathscr{A}$,
$$\mu \Bigg(\bigcup_{i=1}^\infty A_i\Bigg)=\sum_{i=1}^\infty \mu(A_i).$$
Note: The triplet $(X,\mathscr{A},\mu)$ is called a measure space and the tuple $(X,\mathscr{A})$ a measurable space.
$\textbf{Measure}$
Let $X$ be a set and $\mathscr{A}$ a $\sigma$-Algebra over $X$. A function $\mu:\mathscr{A}\to[0,\infty]$ is called a measure if it satisfies the following properties:
1) Null empty set: $\mu(\varnothing)=0$
2) Countable additivity (or $\sigma$-additivity): For any infinite sequence $\{A_i\}$ of pairwise disjoint sets in $\mathscr{A}$,
$$\mu \Bigg(\bigcup_{i=1}^\infty A_i\Bigg)=\sum_{i=1}^\infty \mu(A_i).$$
Note: The triplet $(X,\mathscr{A},\mu)$ is called a measure space and the tuple $(X,\mathscr{A})$ a measurable space.
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๐๐๐๐ฌ๐ฎ๐ซ๐
Let ๐ be a set and ๐ a ๐-Algebra over ๐. A function ๐:๐โ[0,โ] is called a measure if it satisfies the following properties:
1) Null empty set: ๐(โ )=0
2) Countable additivity (or ๐-additivity): For any infinite sequence {๐ดแตข} of pairwise disjoint sets in ๐,
๐(โแตขโโโแชฒ ๐ดแตข)=โแตขโโโแชฒ ๐(๐ดแตข).
Note: The triplet (๐,๐,๐) is called a measure space and the tuple (๐,๐) a measurable space.