Measure theory topics and concepts
| Front | Completion of a measure space |
| Back | The completion of measure space is the refinement of [$](X, B, \mu)[/$] which is is complete. It is denoted [$](X, \overline{B}, \overline{\mu})[/$] where the new sigma algebra consists of sets which differ from B by a sub-null set. |
| Front | Dominated convergence |
| Back | If [$](X, B, \mu)[/$] is a measure space and [$]E_1, E_2, \dots [/$] is a sequence of measurable sets which converge to another set [$]E[/$] in the sense that [$]1_{E_n}[/$] converges pointwise to [$]1_E[/$], we say that the sequence has dominated convergence and has the properties:1. [$]E[/$] is also measurable2. If there exists a measurable set [$]F[/$] of finite measure which contains all of the sets in the sequence then [$]\lim_{n\rightarrow \infty}\mu(E_n) = \mu(E_n)[/$]. |
| Front | Properties of Dirac measure |
| Back | finite additivity |