1-)if $\displaystyle E_1,E_2$ is measurable then show that

$\displaystyle m(E_1 \cup E_2)+m(E_1 \cap E_2)=mE_1+mE_2$

2-)if f(x) is nonnegatif and integrable function than show that

m{$\displaystyle {x:x \in{R},f(x)=\infty}$}=0

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- Dec 17th 2008, 09:02 AMsah_matreel analysis-help(added some more questions)
1-)if $\displaystyle E_1,E_2$ is measurable then show that

$\displaystyle m(E_1 \cup E_2)+m(E_1 \cap E_2)=mE_1+mE_2$

2-)if f(x) is nonnegatif and integrable function than show that

m{$\displaystyle {x:x \in{R},f(x)=\infty}$}=0 - Dec 18th 2008, 03:12 AMMoo
Hello,

Remember that a measure is $\displaystyle \sigma$-additive : if $\displaystyle A \cap B=\emptyset$, then $\displaystyle m(A \cup B)=m(A)+m(B)$

**1.**$\displaystyle m(E_1)=m(E_1 \backslash E_2)+m(E_1 \cap E_2)$

Proof :

It is obvious that $\displaystyle (E_1 \backslash E_2) \cap (E_1 \cap E_2)=\emptyset$ and note that $\displaystyle (E_1 \backslash E_2) \cup (E_1 \cap E_2)=E_1$

So from $\displaystyle \sigma$-additivity, we've proved that.

**2.**

$\displaystyle m(E_1 \cap E_2)+m(E_1 \cup E_2)=m(E_1)+m(E_2)$

Proof :

$\displaystyle E_1 \backslash E_2$, $\displaystyle E_2 \backslash E_1$ and $\displaystyle E_1 \cap E_2$ are disjoint 2 by 2.

And their union is $\displaystyle E_1 \cup E_2$

So by $\displaystyle \sigma$-additivity, $\displaystyle m(E_1 \cup E_2)=m(E_1 \backslash E_2)+m(E_1 \cap E_2)+m(E_2 \backslash E_1)$

Add $\displaystyle m(E_1 \cap E_2)$ to each side of the equality :

$\displaystyle m(E_1 \cap E_2)+m(E_1 \cup E_2)=[m(E_1 \backslash E_2)+m(E_1 \cap E_2)]+[m(E_2 \backslash E_1)+m(E_1 \cap E_2)]$

By**1.**:

$\displaystyle m(E_1 \backslash E_2)+m(E_1 \cap E_2)=m(E_1)$

and $\displaystyle m(E_2 \backslash E_1)+m(E_1 \cap E_2)=m(E_2)$

You're done. - Dec 18th 2008, 06:23 AMsah_mat
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. - Dec 21st 2008, 03:39 AMsah_matsome more questions ı want to ask
question1)

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

question2)

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

question3)

if the set of reel number {$\displaystyle E_n$} n=1 to $\displaystyle \infty$ is disjoint two by two,then construct such sets {$\displaystyle F_n$} n=1 to $\displaystyle \infty$

which holds $\displaystyle \bigcup_{n=1-->\infty}E_n=\bigcup_{n=1-->\infty}F_n$

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.(Bow) - Dec 24th 2008, 02:50 PMsah_mat
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.