I have been given the following question: Let $X = [0, 1]$ and let µ be Lebesgue measure on X. Let $f(x) = x^2$ . Describe the measure $f∗µ$ ( the pushforward)
a.) by calculating $(f∗µ)([a, b])$ for every interval $[a, b] ⊂ R$ b.) by giving the density of $f∗µ$ with respect to Lebesgue measure.
I tried to solve it , especially part a but for b, I do not undertand how to use definition of density.
for part a I said $(f∗µ)([a,b])$ = 0 if $[a,b]∩[0,1] = {c}$ ˝if the intersection is a single point˝ since lesbegue measure of a point is zero. or $(f∗µ)([a,b])$= $\sqrt{d}-\sqrt{c} \, \, if [a,b]∩[0,1]=[c,d]$
but for part be I am trying to use Definition of density of measure: Definition 12 (density). If $ν(A) = \int_{A}^{} f dµ$ , we say that the measure $ν$ has density $f$ with respect to $µ$. should I differentiate $(f∗µ)([a,b])$ in a? any help will be appreciated
Hint: To find the density, it is enough to find $g$ such that $\int_0^x g d\mu =(f*\mu)([0,x])=\sqrt{x}$ for all $x \in [0,1]$, this is because of the fact, that the family of sets $\{[0,x] \:|\: x \in [0,1]\}$ generates the borel sigma algebra on $[0,1]$.