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<div><span class="kw">theorem </span><a NAME="T83"><span class="comment"><font color="firebrick">:: MEASUR11:83</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b3">M</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S</font> holds <br/> ( <font color="Olive" title="b3">M</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  iff  ex <font color="Olive" title="b4">F</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Olive" title="b2">S</font> st <br/>( <font color="Olive" title="b4">F</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Olive" title="b3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> ) ) <span class="kw">by</span> <span class="lab"><a class="txt" href="measur11.html#E98"><span class="lab"><font color="Green" title="E75">LM0902a</font></span></a>, <a class="txt" href="measur11.html#E99"><span class="lab"><font color="Green" title="E76">LM0902b</font></span></a></span>;<br/></div></div>
