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
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
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.
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.
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.