User:Kcbetancourt/Analysis

analysis homework

20. Show that the sum and product of two simple functions are simple. Show that:

$$ \chi _{A\cap B} = \chi _A \cdot \chi _B $$

$$ \chi _{A\cup B} = \chi _A + \chi _B - \chi _A \cdot \chi _B $$

$$ \chi _{A^c} = 1 - \chi _A $$.

$$ \chi _{A\cap B} = \chi _A \cdot \chi _B $$

$$ \chi _{A\cap B}(x) = \begin{cases} 1, & x\in A\cap B \\ 0, & x\notin A\cap B \end{cases} \ $$

Let $$ x\in A\cap B $$

$$ \chi _{A\cap B}(x) = 1 \iff x\in A $$ and $$ x\in B $$

$$ x\in A \iff \chi _A(x) = 1 $$

$$ x\in B \iff \chi _B(x) = 1 $$

So, $$ \chi _{A\cap B}(x) = 1 = 1 \cdot 1 = \chi _A(x) \cdot \chi _B(x) $$

Therefore, when $$ x\in A\cap B $$, $$ \chi _{A\cap B} = \chi _A \cdot \chi _B $$.

Now let $$ x\notin A\cap B $$

$$ \chi _{A\cap B}(x) = 0 \iff x\in A\setminus B $$ or $$ x\in B\setminus A $$

23. Prove Proposition 22 by establishing the following lemmas:

Proposition 22: Let $$ f\ $$ be a measurable function defined on an interval $$ [a,b]\ $$, and assume that $$ f\ $$ takes the values $$ \pm \infty $$ only on a set of measure zero. Then given $$ \varepsilon >0\ $$, we can find a step function $$ g\ $$ and a continuous function $$ h\ $$ such that $$ \left|{f-g}\right| < \varepsilon $$ and $$ \left|{f-h}\right| < \varepsilon $$ except on a set of measure less than $$ \varepsilon $$; i.e., $$ m \left\{{ x: \left|{f(x)-g(x)}\right| \ge \varepsilon }\right\} < \varepsilon $$ and $$ m \left\{{ x: \left|{f(x)-h(x)}\right| \ge \varepsilon }\right\} < \varepsilon $$. If in addition $$ m\le f\le M $$, then we may choose the functions $$ g\ $$ and $$ h\ $$ so that $$ m\le g\le M $$ and $$ m\le h\le M $$.

a.) Given a measurable function $$ f\ $$ on $$ [a,b]\ $$ that takes the values $$ \pm \infty $$ only on a set of measure zero, and given $$ \varepsilon > 0 $$, there is an $$ M\ $$ such that $$ \left|{f}\right| \le M $$ except on a set of measure less than $$ \frac{\varepsilon}{3} $$.

b.) Let $$ f\ $$ be a measurable function on $$ [a,b]\ $$. Given $$ \varepsilon > 0 $$ and $$ M\ $$, there is a simple function $$ \varphi $$ such that $$ \left|{f(x)-\varphi (x)}\right| < \varepsilon $$ except where $$ \left|{f(x)}\right| \ge M $$. If $$ m\le f\le M $$, then we may take $$ \varphi $$ so that $$ m\le \varphi \le M $$.

c.) Given a simple function $$ \varphi $$ on $$ [a,b]\ $$, there is a step function $$ g\ $$ on $$ [a,b]\ $$ such that $$ g(x) = \varphi (x) $$ except on a set of measure less than $$ \frac{\varepsilon}{3} $$. If $$ m\ge \varphi \ge M $$, then we can take $$ g\ $$ so that $$ m\ge g\ge M $$.

d.) Given a step function $$ g $$ on $$ [a,b] $$, there is a continuous function $$ h $$ such that $$ g(x) = h(x) $$ except on a set of measure less than $$ \frac{\varepsilon}{3} $$. If $$ m\ge g\ge M $$, then we may take $$ h $$ so that $$ m\ge h\ge M $$.

24. Let $$ f $$ be measurable and $$ B $$ a Borel set. Then $$ f^{-1} [B] $$ is a measurable set. (The class of sets for which $$ f^{-1} [E] $$ is measurable is a $$ \sigma $$-algebra.)

25. Show that if $$ f $$ is a measurable real-valued function and $$ g $$ a continuous function defined on $$ (-\infty, \infty ) $$, then $$ g \circ f $$ is measurable.

28. Let $$ f_1 $$ be the Cantor ternary function, and define $$ f $$ by $$ f(x) = f_1(x) + x $$.

a.) Show that $$ f $$ is a homeomorphism of $$ [0,1] $$ onto $$ [0,2] $$.

b.) Show that $$ f $$ maps the Cantor set onto a set $$ F $$ of measure 1.

c.) Let $$ g = f^{-1} $$. Show that there is a measurable set $$ A $$ such that $$ g^{-1}[A] $$ is not measurable.

d.) Give an example of a continuous function $$ g $$ and a measurable function $$ h $$ such that $$ h \circ g $$ is not measurable. Compare with Problems 25 and 26.