reel analysis-help

**sah_mat** · December 17th 2008, 09:02 AM

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

**Moo** · December 18th 2008, 03:12 AM

Hello,

Originally Posted by sah_mat

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.

**sah_mat** · December 18th 2008, 06:23 AM

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.

**sah_mat** · December 21st 2008, 03:39 AM

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.

**sah_mat** · December 24th 2008, 02:50 PM

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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