Show that if A is measurable set then the set 3A : = { 3a : a Є A} is measurable. Find its measure. How about the set -3A? Give proofs to your answers. [20 marks]

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- Aug 22nd 2008, 02:34 PMcynthiacheokreal analysis...pls help me prove this!
Show that if A is measurable set then the set 3A : = { 3a : a Є A} is measurable. Find its measure. How about the set -3A? Give proofs to your answers. [20 marks]

- Aug 22nd 2008, 02:44 PMThePerfectHacker
I am not sure how you are defining "measurable", I will assume you mean Jordan measurable.

Also where is $\displaystyle A$ a subset of? It will be assumed that $\displaystyle \emptyset \not = A\subset \mathbb{R}$.

Let $\displaystyle x_0 \in \partial A$ show that $\displaystyle 3x_0 \in \partial (3A)$ and conversely.

Let $\displaystyle \epsilon > 0$.

Since $\displaystyle A$ is measurable it means $\displaystyle \partial A$ can be covered by intervals $\displaystyle I_1, ... ,I_k$ so that $\displaystyle \Sigma_{i=1}^k \text{length}(I_i) < \epsilon/3$.

If $\displaystyle I_i = [a_i,b_i]$ define $\displaystyle J_i = [3a_i,3b_i]$.

Then $\displaystyle J_1,...,J_k$ cover $\displaystyle 3A$ and $\displaystyle \Sigma_{i=1}^k \text{length}(J_i) < \epsilon$.

Thus, $\displaystyle 3A$ is measurable. - Aug 25th 2008, 03:00 AMcynthiacheok
thanks,but wat about the measure of it?The measure we are talking about here is lebesgue measure

- Aug 25th 2008, 09:37 AMmathgeek777