Let A be a measurable subset of [0,1] such that |A|&gt;0, i.e., A has positive measure. Prove that the set cos ⁡ ( A ) = { cos ⁡ x : x ∈ A } has strictly smaller measure than A.The cosine graph tells us that the function is one-one at [ 0 , 1 ] and clearly the measure of [ cos ⁡ 1 , 1 ] is smaller than 1. But other than that I have no idea how to even begin. Any help would be appreciated.

Question

Let   A be a measurable subset of [0,1] such that |A|&amp;gt;0, i.e., A has positive measure. Prove that the set   cos  ⁡  (  A  )  =  {  cos  ⁡  x  :  x  ∈  A  } has strictly smaller measure than   A.The cosine graph tells us that the function is one-one at   [  0  ,  1  ] and clearly the measure of   [  cos  ⁡  1  ,  1  ] is smaller than 1. But other than that I have no idea how to even begin. Any help would be appreciated.

Giovani Hickman · Accepted Answer

∫              cos        ⁡        (        1        )                    cos        ⁡        (        0        )              f    (    t    )        d    t    =          ∫      0      1        f    (    cos    ⁡    (    x    )    )    sin    ⁡    (    x    )        d    x  where you can regard   f  (  t  ) as the indicator function of   cos  ⁡  (  A  ).In this way, since   cos  ⁡  (  x  ) is a bijection from [0,1] to   [  cos  ⁡  (  1  )  ,  cos  ⁡  (  0  )  ],we have that   f  (  cos  ⁡  (  x  )  ) is the indicator function of   A.Now, how large is   sin  ⁡  (  x  ) for   0  ≤  x  ≤  1?

Let A be a measurable subset of [0,1] such that |A|>0, i.e., A has positive measure. Prove that

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