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Math Help - real analysis...pls help me prove this!

  1. #1
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    real 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]

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  2. #2
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    Quote Originally Posted by cynthiacheok View Post
    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]
    I am not sure how you are defining "measurable", I will assume you mean Jordan measurable.
    Also where is A a subset of? It will be assumed that \emptyset \not = A\subset \mathbb{R}.
    Let x_0 \in \partial A show that 3x_0 \in \partial (3A) and conversely.
    Let \epsilon > 0.
    Since A is measurable it means \partial A can be covered by intervals I_1, ... ,I_k so that \Sigma_{i=1}^k \text{length}(I_i) < \epsilon/3.
    If I_i = [a_i,b_i] define J_i = [3a_i,3b_i].
    Then J_1,...,J_k cover 3A and \Sigma_{i=1}^k \text{length}(J_i) < \epsilon.
    Thus, 3A is measurable.
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  3. #3
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    thanks,but wat about the measure of it?The measure we are talking about here is lebesgue measure
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  4. #4
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    Quote Originally Posted by cynthiacheok View Post
    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]
    Is that a question on a quiz or test? It looks like one....
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