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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <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/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:92"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">f</font> <a href="mesfunc6.html#R2" target="_self">is_simple_func_in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 ;<br/>

<a NAME="E2:92"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc5.html#T43">MESFUNC5:43</a></i>;<br/>
<a NAME="E4:92"/>
 <a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#T37">MESFUNC2:37</a></i>;<br/>
<a NAME="E5:92"/><b>then </b>
( <font color="Maroon">f</font> <a href="mesfunc6.html#R1" target="_self">is_measurable_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:92"/>
 <a href="mesfunc6.html#K6" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K6">integral+</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/>

<b>hence </b><a NAME="E7:92"/>
 <a href="mesfunc6.html#K6" target="_self">Integral</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc5.html#K5">integral'</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T83">MESFUNC5:83</a></i>;<br/>


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