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Math Help - measurable sets,

  1. #1
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    measurable sets,

    Hi,

    I am struggling to provide sound mathematical proofs for these. Intuitively it's pretty straight forward.

    Prove that the following sets is measurable and has zero area.
    1) A set of a finite number of points in a plane.
    2) The union of a finite collection of line segments in a plane?

    Each finite line segment has width zero, so area zero. A point has length zero and height zero so it's zero.

    Given
    <br />
given: h \ge 0, and,  k \ge 0<br />
    <br />
rectangles\ are\ sets\ such\ that\ : (x,y)\mid \{x \le k, y \le h\}<br />

    Therefore we can say that for each line k is zero, and h is the length of the line, therefore the area is zero.

    For each point h and k are zero, and therefore the area is zero.

    Why are the set's measurable though?

    How do we know mathematically there is one number c (zero) such that
    <br />
a(S) \le c \le a(T)<br />

    or is this trivially demonstrated by the fact that h or k are zero above?
    I want to be sure I understand this concept properly.

    Thanks
    Regards
    Craig.
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  2. #2
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    Thanks
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    Quote Originally Posted by craigmain View Post
    Hi,

    I am struggling to provide sound mathematical proofs for these. Intuitively it's pretty straight forward.

    Prove that the following sets is measurable and has zero area.
    1) A set of a finite number of points in a plane.


    Let X:=\{x_1,...,x_n\}\subset\mathbb{R} be our finite set, and choose any \epsilon >0

    Then \displaystyle{\left\{\left(x_i-\frac{\epsilon}{3n},\,x_i+\frac{\epsilon}{3n}\righ  t)\right\}_{i=1,2,...,n} is a cover of the

    set X with length <\epsilon\Longrightarrow by definition, its measure is zero.

    Try now to mimic the above for a finite union of finite sets in the plane, this time taking (open, say) rectangles

    instead of open intervals...

    Tonio



    2) The union of a finite collection of line segments in a plane?

    Each finite line segment has width zero, so area zero. A point has length zero and height zero so it's zero.

    Given
    <br />
given: h \ge 0, and,  k \ge 0<br />
    <br />
rectangles\ are\ sets\ such\ that\ : (x,y)\mid \{x \le k, y \le h\}<br />

    Therefore we can say that for each line k is zero, and h is the length of the line, therefore the area is zero.

    For each point h and k are zero, and therefore the area is zero.

    Why are the set's measurable though?

    How do we know mathematically there is one number c (zero) such that
    <br />
a(S) \le c \le a(T)<br />

    or is this trivially demonstrated by the fact that h or k are zero above?
    I want to be sure I understand this concept properly.

    Thanks
    Regards
    Craig.
    .
    Follow Math Help Forum on Facebook and Google+

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