<?xml version="1.0"?>
<div><span class="kw">theorem </span><a NAME="T44"><span class="comment"><font color="firebrick">:: MESFUN15:44</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b4">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Olive" title="b1">f</font> <a href="integr24.html#R3" title="INTEGR24:pred.3">is_improper_integrable_on</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">A</font> is  <a href="mesfunc5.html#V2" title="MESFUNC5:attr.2">nonpositive</a>  holds <br/>(  <a href="integr24.html#K3" title="INTEGR24:func.3">improper_integral</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">A</font></span>)</span>) &amp; (  ex <font color="Olive" title="b5">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b5">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">f</font> <a href="integr10.html#R2" title="INTEGR10:pred.2">is_left_ext_Riemann_integrable_on</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b1">f</font> <a href="integr10.html#R1" title="INTEGR10:pred.1">is_right_ext_Riemann_integrable_on</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b3">b</font> ) implies <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">A</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a>  <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>  ) &amp; ( (  for <font color="Olive" title="b5">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b5">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">f</font> <a href="integr10.html#R2" title="INTEGR10:pred.2">is_left_ext_Riemann_integrable_on</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">c</font> holds <br/> not <font color="Olive" title="b1">f</font> <a href="integr10.html#R1" title="INTEGR10:pred.1">is_right_ext_Riemann_integrable_on</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b3">b</font> ) implies  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">A</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )</div></div>
