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Math Help - Measure theory

  1. #1
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    Measure theory

    1.For k>0 and A⊂R,let kA={kx:x∈A}.Show that m^* (kA)=km^* (A) and A is measurable if and only if kA is measurable.

    2.For A⊂R let A={-x:x∈A}.Show that m^* (A)=m^* (-A) and A is measurable if and only if -A is measurable.

    3.Show that the sequence {f_n} where f_n=nxe^(-nx^2 ) for n=1,2,3, converges pointwise in [0,1].Is the convergence uniform?Justify.
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  2. #2
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    Re: Measure theory

    You should specify what the exact measure space is. I assume it's the Lebesgue measure on the Lebesgue measurable sets of \mathbb{R}.

    For #1, assuming A is measurebale, use the outer measure definition. Observe that if a "chunk of A" that's covered by an interval [a, b], then the corresponding "chunk of kA" is covered by the interval [ka, kb]. Likewise, if a "chunk of kA" is covered by the interval [a, b], then the corresponding "chunk of A" is covered by the interval [a/k, b/k]. The k=0 special case is easy, and you should also pay attention to what happens when m(A) is infinite.

    If you understood #1, then #2 uses a similar idea.

    How far have you gotten on #3? Have you determined the pointwise convergence? Do you know what uniform convergence means?
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