Thread: 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.

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

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?