Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$T: [0,1)^{2}\rightarrow[0,1)^{2}$ by

$T(x,y) = (2x,\frac{y}{2})$, with $0 \leq x < \frac{1}{2}$

and

$T(x,y) = (2x-1, \frac{y+1}{2})$, with $\frac{1}{2} \leq x < 1$

In class we said this $T$ is

a) invertivle

b) measurable

c) measure preserving.

My last Analysis and Stochasitc class is quite a time ago and so I do not see that easy how $T$ fulfills a)-c)....

I hope someone can tell my why this is true, or give me some hints how I can find that out myseld :)

Best, Luca

share|improve this question
    
To check it's invertible, you could simply find the inverse. It's a piecewise linear map so it shouldn't be too bad. You know it's measurable because it's piece-wise continuous. You should check the measure-preserving property by drawing what happens to points under $T$. –  Christopher A. Wong Oct 9 '12 at 23:37

1 Answer 1

youre taking the square, smooshing it down and stretching it out. then cut off the excess and put it on top. (draw a picture). clearly invertible (just undo what you did). preserves measure (doesnt change areas, should be clear, what happens to the area of a little rectangle under $T^{-1}$?).

share|improve this answer
    
Hey! First thanks for the Answers! My $T^{-1}(x,y) = (\frac{x}{2},2y)$ for $0 \leq y < \frac{1}{2}$ And $= (\frac{x+1}{2},2y-1)$ for $\frac{1}{2} \leq y < 1$. –  Luca Oct 10 '12 at 0:40
    
I pushed enter too early ;) Hey! First thanks for the Answers! The baker trafo describes taking a rectangle, strech it, fold it and turn it a quater. Hence if I name the two halfs $A$ and $B$, then $A$ has $\frac{1}{2}$ of the area and $T(A)$ sends it again to $\frac{1}{2}$ of the area. And then $T(A)$ intersects $A$ and $B$ in a square of area $\frac{1}{2}$. Well so informally, if we take the measure of $A$, and go the streching/folding.. back, than we still have the same area for $A$? How do I write this more formally? –  Luca Oct 10 '12 at 0:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.