Editing Monotone convergence theorem (section)

====Proof (lemma 2)====
Write 
<math>s=\sum^n_{k=1}c_k\cdot {\mathbf 1}_{A_k},</math>
with <math>c_k\in{\mathbb R}_{\geq 0}</math> and measurable sets <math>A_k\in\Sigma</math>.  Then 
:<math>\nu_s(A)=\sum_{k =1}^n c_k \mu(A\cap A_k).</math>
Since finite positive linear combinations of countably additive set functions are countably additive, to prove countable additivity of <math>\nu_s</math>  it suffices to prove that, the set function defined by <math>\nu_B(A) = \mu(B \cap A)</math> is countably additive for all <math>A \in \Sigma</math>. But this follows directly from the countable additivity of <math>\mu</math>.

=====Continuity from below=====

'''Lemma 3.''' Let <math>\mu</math> be a measure, and <math>A = \bigcup^\infty_{i=1}A_i</math>, where
:<math>
A_1\subseteq\cdots\subseteq A_i\subseteq A_{i+1}\subseteq\cdots\subseteq A
</math>
is a non-decreasing chain with all its sets <math>\mu</math>-measurable. Then
:<math>\mu(A)=\sup_i\mu(A_i).</math>