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<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/>



<a NAME="E1:29"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">A</font> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font> implies for <font color="Olive">M</font> being   <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font> holds  <font color="Maroon">A</font> <a href="measure1.html#R2">is_measurable</a> <font color="Olive">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E2:29_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">A</font> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:29"/>
( ( for <font color="Olive">M</font> being   <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font> holds  <font color="Maroon">A</font> <a href="measure1.html#R2">is_measurable</a> <font color="Olive">M</font> ) implies <font color="Maroon">A</font> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">M</font> being   <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font> holds  <font color="Maroon">A</font> <a href="measure1.html#R2">is_measurable</a> <font color="Olive">M</font>
 ;<br/>

<b>consider </b><font color="Maroon">M</font> being    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E3:29_2"/>
<font color="Maroon">A</font> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:29_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E5:29_2"/>
<font color="Maroon">A</font> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
( <font color="Maroon">A</font> <a href="mesfunc1.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">S</font> iff for <font color="Olive">M</font> being   <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font> holds  <font color="Maroon">A</font> <a href="measure1.html#R2">is_measurable</a> <font color="Olive">M</font> )
 <b>by </b><i/>;<br/>


</div>
