MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document I participate the stochastic course and we now speak about summable families. There we have the following definition:Let &#x03A9; be countable and a : &#x03A9; &#x2192; R + &#x222A; { &#x221E; } be a map. Then we define &#x2211; &#x03A9; a ( &#x03C9; ) := sup F &#x2282; &#x03A9; , | F | &lt; &#x221E; &#x2211; F a ( &#x03C9; )Now our Prof said that we can consider &#x2211; &#x03A9; a ( &#x03C9; ) as the integral of math xmlns="http://www.w3.org/1998/Math/MathML"> a over &#x03A9; to get a better connetion to measure theory afterwards when we speak about Fatou&#39;s lemma, Beppo Levi theorem ect. Because all this theorems we have seen with integrals last semester.But somehow I don&#39;t see why this sum is equal to the integral. So I know from measure theory that if we have a simple function f : &#x03A9; &#x2192; R + &#x222A; { &#x221E; } where f ( &#x03A9; ) = { b 1 , . . . , b n } finately many then &#x222B; &#x03A9; f &#xA0; d &#x03BC; = &#x2211; i = 1 n f ( b i ) &#x22C5; &#x03BC; ( &#x03A9; i )where &#x03A9; i = f &#x2212; 1 ( { b i } ). So but here I don&#39;t think that this has to do something with simple functions right?Therefore I wanted to ask you if someone could explain me why we can see this sum as the integral of a over &#x03A9;.

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MathJax(?): Can&#039;t find handler for document MathJax(?): Can&#039;t find handler for document I participate the stochastic course and we now speak about summable families. There we have the following definition:Let   &amp;#x03A9; be countable and   a  :  &amp;#x03A9;  &amp;#x2192;            R        +    &amp;#x222A;  {  &amp;#x221E;  } be a map. Then we define      &amp;#x2211;    &amp;#x03A9;    a  (  &amp;#x03C9;  )  :=      sup          F      &amp;#x2282;      &amp;#x03A9;      ,              |            F              |            &amp;lt;      &amp;#x221E;            &amp;#x2211;    F    a  (  &amp;#x03C9;  )Now our Prof said that we can consider       &amp;#x2211;    &amp;#x03A9;    a  (  &amp;#x03C9;  ) as the integral of math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;  a over   &amp;#x03A9; to get a better connetion to measure theory afterwards when we speak about Fatou&amp;#39;s lemma, Beppo Levi theorem ect. Because all this theorems we have seen with integrals last semester.But somehow I don&amp;#39;t see why this sum is equal to the integral. So I know from measure theory that if we have a simple function   f  :  &amp;#x03A9;  &amp;#x2192;            R        +    &amp;#x222A;  {  &amp;#x221E;  } where   f  (  &amp;#x03A9;  )  =  {      b    1    ,  .  .  .  ,      b    n    } finately many then      &amp;#x222B;    &amp;#x03A9;    f  &amp;#xA0;      d    &amp;#x03BC;  =      &amp;#x2211;          i      =      1        n    f  (      b    i    )  &amp;#x22C5;  &amp;#x03BC;  (      &amp;#x03A9;    i    )where       &amp;#x03A9;    i    =      f          &amp;#x2212;      1        (  {      b    i    }  ). So but here I don&amp;#39;t think that this has to do something with simple functions right?Therefore I wanted to ask you if someone could explain me why we can see this sum as the integral of   a over   &amp;#x03A9;.

Sasha Pacheco · Accepted Answer

First prove that if   Ω  =  {      ω    1    ,      ω    2    ,  …  }, then       ∑          ω      ∈      Ω        a  (  ω  )  =      ∑          i      =      1              ∞        a  (      ω    i    ).Now take   μ to be the counting measure on   Ω. Really, without loss of generality, you can assume   Ω  =      N  , so you want to prove that       ∫                  N              a    d  μ  =      ∑          i      =      1              ∞            a    i  . This is obvious if there exists   N such that       a    i    =  0 for   i  ≥  N. The general case follows from monotone convergence theorem.

I participate the stochastic course and we now speak about summable families. There we have the foll

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