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Math Help - reel analysis-help

  1. #1
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    reel analysis-help(added some more questions)

    1-)if E_1,E_2 is measurable then show that
    m(E_1 \cup E_2)+m(E_1 \cap E_2)=mE_1+mE_2
    2-)if f(x) is nonnegatif and integrable function than show that
    m{ {x:x \in{R},f(x)=\infty}}=0
    Last edited by sah_mat; December 21st 2008 at 04:41 AM.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by sah_mat View Post
    1-)if E_1,E_2 is measurable then show that
    m(E_1 \cup E_2)+m(E_1 \cap E_2)=mE_1+mE_2
    Remember that a measure is \sigma-additive : if A \cap B=\emptyset, then m(A \cup B)=m(A)+m(B)

    1. m(E_1)=m(E_1 \backslash E_2)+m(E_1 \cap E_2)
    Proof :
    It is obvious that (E_1 \backslash E_2) \cap (E_1 \cap E_2)=\emptyset and note that (E_1 \backslash E_2) \cup (E_1 \cap E_2)=E_1
    So from \sigma-additivity, we've proved that.


    2.
    m(E_1 \cap E_2)+m(E_1 \cup E_2)=m(E_1)+m(E_2)
    Proof :
    E_1 \backslash E_2, E_2 \backslash E_1 and E_1 \cap E_2 are disjoint 2 by 2.
    And their union is E_1 \cup E_2
    So by \sigma-additivity, m(E_1 \cup E_2)=m(E_1 \backslash E_2)+m(E_1 \cap E_2)+m(E_2 \backslash E_1)

    Add m(E_1 \cap E_2) to each side of the equality :
    m(E_1 \cap E_2)+m(E_1 \cup E_2)=[m(E_1 \backslash E_2)+m(E_1 \cap E_2)]+[m(E_2 \backslash E_1)+m(E_1 \cap E_2)]

    By 1. :
    m(E_1 \backslash E_2)+m(E_1 \cap E_2)=m(E_1)
    and m(E_2 \backslash E_1)+m(E_1 \cap E_2)=m(E_2)

    You're done.
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  3. #3
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    thans a lot mr.moo,what do u think about the second question,i think it is a lemma but ı could not find it.i hope some of my friends here help my questions,becouse these are may be my exam questions at next week,and my last chance to pass reel analysis.
    i appreciate your helps.
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  4. #4
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    some more questions ı want to ask

    question1)
    if f(x) is nonnegative and integrable funciton at (-\infty,\infty) then prove F(x)=\int_{(-\infty,x]}f(y)dy is continuous.


    question2)
    if f(x) is integrable funciton at [a,b] then prove the function g(x)=\int_{[a,b]}f(t)dt is continuous at [a,b]


    question3)
    if the set of reel number { E_n} n=1 to \infty is disjoint two by two,then construct such sets { F_n} n=1 to \infty
    which holds \bigcup_{n=1-->\infty}E_n=\bigcup_{n=1-->\infty}F_n

    i have already been thankfull for helps ı am trying to solve other questions .i wish the citizens of math help forum are going to help me.
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  5. #5
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    ı have 6 hours to my make up exam but i can not solve this questions,if there is someone hearing my plea come on anad help meeeeee!!!i am still studying i will check with all my hope after 4 hours.thanks a lot.
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