User:MCPOliseno /Math710 Essay

1 Measure

(a) Summarize the notion of Lebesgue outer measure and the role of Caratheodoryís measurability criterion. You should explain what is gained by restriction of outer measure to the class of measurable sets.

Lebesgue outer measure: m*(A) = inf {$$ \sum \ $$ t $$ (A_n) \ $$}, where the infimum is taken over all countable collections of open intervals $$ (A_n) \ $$, such that $$ A \subset \cup A_n \ $$ and $$ tA_n \ $$ is the standard length of the interval $$ A_n \ $$. Lebesgue outer measure is countably subadditive rather than finitely subaditive, like Jordan outer measure. Since outer measures are countably subadditive, rather than countably additive, they are weaker than measures, but outer measure can measure all subsets of a set, unlike measure which can only measure a $$ \sigma \ $$-algebra of measurable sets. Caratheodoryís measurability criterion is shown by, let $$ \mu \ $$* be an outer measure on a set X. A set E $$ \subset \ $$ X is said to be Caratheodoryís measurable with respect to $$ \mu \ $$* is one has math> \mu \ *(A) = $$ \mu \ $$*(A $$ \cap \ $$ E) + $$ \mu \ $$* (A\E), for every set A $$ \subset \ $$ X.