User:Kcbetancourt/AnalysisHW4

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.