User:J D Bowen/Math710 HW4

3.20) We aim to show $$\chi_{A\cap B}=\chi_A \cdot \chi_B \ $$.


 * Case 1: $$x\not \in A \ \and \ x\not\in B \ $$.

This implies $$x\not \in A \cap B \ $$. We have

$$\chi_{A\cap B}=0 = 0\cdot 0 = \chi_A(x)\cdot \chi_B(x) \ $$.


 * Case 2: $$x\in A \ \and \ x\not\in B \ $$.

This implies $$x\not \in A \cap B \ $$. We have

$$\chi_{A\cap B}=0 = 1\cdot 0 = \chi_A(x)\cdot \chi_B(x) \ $$.


 * Case 3: $$x\not\in A \ \and \ x\in B \ $$.

This implies $$x\not\in A \cap B \ $$. We have

$$\chi_{A\cap B}=0 = 0\cdot 1 = \chi_A(x)\cdot \chi_B(x) \ $$.


 * Case 4: $$x\in A \ \and \ x\in B \ $$.

This implies $$x \in A \cap B \ $$. We have

$$\chi_{A\cap B}=1 = 1\cdot 1 = \chi_A(x)\cdot \chi_B(x) \ $$.

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23) We aim to show the following 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 $$ \epsilon >0\ $$, we can find a step function $$ g\ $$ and a continuous function $$ h\ $$ such that $$ \left|{f-g}\right| < \epsilon $$ and $$ \left|{f-h}\right| < \epsilon $$ except on a set of measure less than $$ \epsilon $$; i.e., $$ m \left\{{ x: \left|{f(x)-g(x)}\right| \ge \epsilon }\right\} < \epsilon $$ and $$ m \left\{{ x: \left|{f(x)-h(x)}\right| \ge \epsilon }\right\} < \varepsilon $$. 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 $$ \epsilon > 0 $$, there is an $$ M\ $$ such that $$ \left|{f}\right| \le M $$ except on a set of measure less than $$ \frac{\epsilon}{3} $$.

b.) Let $$ f\ $$ be a measurable function on $$ [a,b]\ $$. Given $$ \epsilon > 0 $$ and $$ M\ $$, there is a simple function $$ \varphi $$ such that $$ \left|{f(x)-\varphi (x)}\right| < \epsilon $$ 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 $$ \frac{\epsilon}{3} $$. 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 $$ \frac{\epsilon}{3} $$. If $$ m\ge g\ge M $$, then we may take $$ h $$ so that $$ m\ge h\ge M $$.

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3.24) We aim to show that if $$ f \ $$ is measurable and $$ B \ $$ is a Borel set, then $$ f^{-1} (B) \ $$ is a measurable set - that is to say, the class of sets for which $$ f^{-1} (E) \ $$ is measurable is a $$ \sigma \ $$-algebra.)

Let $$C \ $$ be the collection of sets for which $$ f^{-1} (E) \ $$ is measurable.


 * 1) Closed under complementation:
 * Suppose $$E\in C \ $$. Then we have $$f^{-1}(\overline{E})=\overline{f^{-1}(E)} \ $$, which is the complement of a measurable set and hence measurable, and so $$\overline{E}\in C \ $$.


 * 2) Closed under countable unions:
 * Let $$E_j, \ j=1, \dots \ $$ be a countable collection of sets in $$C \ $$. Then


 * $$f^{-1}\left({\bigcup_{j=1}^\infty E_j }\right) = \bigcup_{j=1}^\infty f^{-1}(E_j) \ $$, which is a countable union of measurable sense and hence measurable. So $$\cup_{j=1}^\infty E_j \in C $$.

Hence, $$C \ $$ is a $$\sigma \ $$-algebra.

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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.