Okay, let ( X , M , μ ) , ( Y , N , ν ) be measure spaces. Show for characteristic functions χ A ∈ L + ( X ) , χ B ∈ L + ( Y ), that χ A × B ∈ L + ( X × Y ) and ∫ χ A × B d ( μ × ν ) = ∫ χ A d μ ∫ χ B d νAttempt: By definition of the product measure, ∫ χ A × B d ( μ × ν ) = μ ( A ) ν ( B ) = ∫ χ A d μ ∫ χ B d ν. Since ∫ χ A d μ &lt; ∞ and ∫ χ B d ν &lt; ∞, we see that ∫ χ A d μ ∫ χ B d ν &lt; ∞ and χ A × B ∈ L + ( X × Y ). That is all that is needed to show. Is this correct?

Question

Okay, let   (  X  ,      M    ,  μ  )  ,  (  Y  ,      N    ,  ν  ) be measure spaces. Show for characteristic functions       χ    A    ∈      L          +        (  X  )  ,      χ    B    ∈      L          +        (  Y  ), that       χ          A      ×      B        ∈      L          +        (  X  ×  Y  ) and  ∫      χ          A      ×      B        d  (  μ  ×  ν  )  =  ∫      χ    A    d  μ  ∫      χ    B    d  νAttempt: By definition of the product measure,   ∫      χ          A      ×      B        d  (  μ  ×  ν  )  =  μ  (  A  )  ν  (  B  )  =  ∫      χ    A    d  μ  ∫      χ    B    d  ν. Since   ∫      χ    A    d  μ  &amp;lt;  ∞ and   ∫      χ    B    d  ν  &amp;lt;  ∞, we see that   ∫      χ    A    d  μ  ∫      χ    B    d  ν  &amp;lt;  ∞ and       χ          A      ×      B        ∈      L          +        (  X  ×  Y  ). That is all that is needed to show. Is this correct?

Jaruckigh · Accepted Answer

It is easy to see from the definition of measurability that for any measurable space   (  Z  ,  F  ),   S  ⊂  Z,       χ    S   is measurable if and only if   S is measurable.If   A  ∈      M   and   B  ∈      N  , then by definition of product measure,   A  ×  B  ∈      M    ⊗      N  . Hence       χ          A      ×      B       is measurable.

Okay, let ( X , <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mat

Answered question

Answer & Explanation

New Questions in Pre-Algebra