Editing Measure (mathematics) (section)

====Results regarding semifinite measures====
<!--Help appreciated, please add results / check the c.l.d. product measure thing-->
* Let <math>\mathbb F</math> be <math>\R</math> or <math>\C,</math> and let <math>T:L_\mathbb{F}^\infty(\mu)\to\left(L_\mathbb{F}^1(\mu)\right)^*:g\mapsto T_g=\left(\int fgd\mu\right)_{f\in L_\mathbb{F}^1(\mu)}.</math> Then <math>\mu</math> is semifinite if and only if <math>T</math> is injective.{{sfn|Fremlin|2016|loc=part (a) of Theorem 243G, p. 159}}{{sfn|Fremlin|2016|loc=Section 243K, p. 162}} (This result has import in the study of the [[Lp space#Dual spaces|dual space of <math>L^1=L_\mathbb{F}^1(\mu)</math>]].)
* Let <math>\mathbb F</math> be <math>\R</math> or <math>\C,</math> and let <math>{\cal T}</math> be the topology of convergence in measure on <math>L_\mathbb{F}^0(\mu).</math> Then <math>\mu</math> is semifinite if and only if <math>{\cal T}</math> is Hausdorff.{{sfn|Fremlin|2016|loc=part (a) of the Theorem in Section 245E, p. 182}}{{sfn|Fremlin|2016|loc=Section 245M, p. 188}}
* (Johnson) Let <math>X</math> be a set, let <math>{\cal A}</math> be a sigma-algebra on <math>X,</math> let <math>\mu</math> be a measure on <math>{\cal A},</math> let <math>Y</math> be a set, let <math>{\cal B}</math> be a sigma-algebra on <math>Y,</math> and let <math>\nu</math> be a measure on <math>{\cal B}.</math> If <math>\mu,\nu</math> are both not a <math>0-\infty</math> measure, then both <math>\mu</math> and <math>\nu</math> are semifinite if and only if [[Product measure|<math>(\mu\times_\text{cld}\nu)</math>]]<math>(A\times B)=\mu(A)\nu(B)</math> for all <math>A\in{\cal A}</math> and <math>B\in{\cal B}.</math> (Here, <math>\mu\times_\text{cld}\nu</math> is the measure defined in Theorem 39.1 in Berberian '65.{{sfn|Berberian|1965|loc=Theorem 39.1, p. 129}}) <!--To check: Is this actually the ''c.l.d. product measure''?{{sfn|Fremlin|2016|loc=Definition 251F, p. 206}}-->