Editing Outer measure (section)

==Restriction and pushforward of an outer measure==
Let <math>\mu</math> be an outer measure on the set <math>X </math>.
===Pushforward===
Given another set <math>Y</math> and a map <math>f:X\to Y </math> define <math>f_\sharp \mu : 2^Y \to [0, \infty]</math> by
:<math>\big(f_\sharp\mu\big)(A)=\mu\big(f^{-1}(A)\big).</math>
One can verify directly from the definitions that <math>f_\sharp \mu</math> is an outer measure on <math>Y</math>.
===Restriction===
Let {{mvar|B}} be a subset of {{mvar|X}}. Define {{math|''μ''<sub>''B''</sub> : 2<sup>''X''</sup>→[0,∞]}} by
:<math>\mu_B(A)=\mu(A\cap B).</math>
One can check directly from the definitions that {{math|''μ''<sub>''B''</sub>}} is another outer measure on {{mvar|X}}.
===Measurability of sets relative to a pushforward or restriction===
If a subset {{mvar|A}} of {{mvar|X}} is {{math|''μ''}}-measurable, then it is also {{math|''μ''<sub>''B''</sub>}}-measurable for any subset {{mvar|B}} of {{mvar|X}}.

Given a map {{math|''f'' : ''X''→''Y''}} and a subset {{mvar|A}} of {{mvar|Y}}, if {{math|''f''<sup> −1</sup>(''A'')}} is {{math|''μ''}}-measurable then {{mvar|A}} is {{math|''f''<sub>#</sub> ''μ''}}-measurable. More generally, {{math|''f''<sup> −1</sup>(''A'')}} is {{math|''μ''}}-measurable if and only if {{mvar|A}} is {{math|''f''<sub>#</sub> (''μ''<sub>''B''</sub>)}}-measurable for every subset {{mvar|B}} of {{mvar|X}}.