Measure

Definition

A function defined on a measurable space $(\Omega ,\mathcal{F})$, in which satisfies the following conditions

• Non-negativity: $\mu :\mathcal{F}\to {\mathbf{R}}_0^{+}$
• $\mu (\varnothing )=0$
• Countable additivity: $\mu (\underset{i}{⨄}{A}_i)=\sum \mu (A_i)$

where $\uplus$ is a union of the pairwise disjoint sets

Kinds

• Probability Measure
• Finite Measure
• Sigma-Finite Measure
• Counting measure
• Dirac measure

probability measure $\subset$ finite measure $\subset$ $\sigma$-finite measure

Summary

	$\mu (\varnothing )=0$	$\sigma$-add	$A_i\uparrow \Omega$, $\mu (A_i)<\infty$	$\mu (\Omega )<\infty$	$\mu (\Omega )=1$		monotone	$\sigma$-subadd	conti-below	conti-above
msr	$O$	$O$	$X$	$X$	$X$		$O$	$O$	$O$	$\Delta$
$\sigma$-finite-msr	$O$	$O$	$O$	$X$	$X$		$O$	$O$	$O$	$\Delta$
finite-msr	$O$	$O$	$O$	$O$	$X$		$O$	$O$	$O$	$O$
prob-msr	$O$	$O$	$O$	$O$	$O$		$O$	$O$	$O$	$O$

Notations

• $\sigma$-additive: $\mu (\underset{i=1}{\overset{\infty }{⨄}}{B}_i)={\sum }_{i=1}^{\infty }\mu (B_i)$
• $\sigma$-subadditive: $\mu (\bigcup _{i=1}^{\infty }{A}_i)\le \sum _{i=1}^{\infty }\mu (A_i)$
• monotone: $A\subset B\Rightarrow \mu (A)\le \mu (B)$
• continuous from below: $A_i\uparrow A\Rightarrow \mu (\underset{n\to \infty }{\lim }{A}_i)=\underset{n\to \infty }{\lim }\mu (A_i)$
• continuous from above: $[A_i\downarrow A\text{and}\mu (A_1)<\infty ]\Rightarrow \mu (\underset{n\to \infty }{\lim }{A}_i)=\underset{n\to \infty }{\lim }\mu (A_i)$