Math Help - reel analysis-help

1. 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

2. 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.

3. 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.

4. 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.

5. 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.