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    Lebesgue Measurable Set

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    Let A be a set in R^n, we denote by A + x_o a parallel shift of A by x_o to A + x_o,
    A + x_o = { x : x = y + x_o, y in A}.

    Now, if A is a lebesgue measurable then show that

    1). x_o + A is also lebesgue measurable

    2). m(A) = m(x_o + A)

    Can someone check my answer and tell me if it is correct or not?

    My work:
    since A is a subset of R^n then A^c ( A compliment) = R^n - A and therefore A^c union A = R^n
    and (A + x_o ) union (A^c + x_o) = R^n
    Since A+x_o in R^n and (A+x_o)^c = A^c + x_o
    ( to show (A + x_o)^c = A^c + x_o :
    let z be element of (A+x_o)^c then z doesn't belong to A + x_o, since x_o is a point then z doesn't belong to x_o, and z doesn't belong to A so z belongs to A^c and therefore z belongs to A^c + x_o
    To do the other way, it is almost the same just reverse the procedure)

    Nowwe get the following
    E = ( E intersection A) union ( E intersection A^c) , where E is a subset of R^n
    and E = (E intersection ( A + x_o) ) union( E intersection ( A^c + x_o))
    so
    (E intersection A) union ( E intersection A^c) = ( E intersection ( A + x_o)) union
    (E intersection ( A^c + x_o))

    so A+x_o is lebesgue measurable

    now for part 2
    m(E intercetion A) union ( E intersection A^c)) =< m((E intersection ( A + x_o)) union( E intersection (A^c + x_o))

    and

    m( E intersection ( A+x_o) union( E intersection (A^c+x_o)) =< m ((E intersection A) union( E intersection A^c))

    so m( (E intersection ( A + x_o) ) union ( E intersection ( A^c + x_o ) ) =
    m(( E intersection A) union( E intersection A^c)

    so m(E) = m ( (E intersection ( A + x_o) ) union ( E intersection ( A^c + x_o))
    = m( ( E intersection A) union ( E intersection A^c))

    © BrainMass Inc. brainmass.com November 24, 2021, 11:50 am ad1c9bdddf
    https://brainmass.com/math/combinatorics/lebesgue-measurable-set-54188

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    Mathematics, Real Variables
    lebesgue measurable set
    ________________________________________
    Let A be a set in R^n, we denote by A + xo a parallel shift of A by xo to A + xo,
    A + xo = { x : x = y + xo, y in A}.

    Now, if A is a lebesgue measurable then show that

    1). xo + A is also lebesgue measurable

    2). m(A) = m(xo + A) ...

    Solution Summary

    Lebesgue Measurable Sets are investigated. The solution is detailed and well presented.

    $2.49

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