Approaching the Measure of a Set: Strategies for Finding f(Eα)

[laurabon]

Homework Statement

let f : [a, b] → R ,
α ≥ 0 and Eα = {x ∈ [a, b] : exists f'(x) e |f'(x)|≤ α}
show that m (f(Eα)) ≤ α m(Eα)

Relevant Equations

.

my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}

 

[fresh_42]

It would help a lot if you used LaTeX. Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/

 

[FactChecker]

Start with an arbitrary $\epsilon \gt 0$. Use the definition of the (outer) measure of a set to find a countable set of open intervals, $C_{epsilon}$ covering $E\alpha$ where the summed length of the intervals is less than $m( E\alpha) + \epsilon$. Use the definition of the derivative to find smaller intervals within each interval, $I_n$, of $C_{epsilon}$ whose image, under $f$ is of length smaller than $\alpha$ length($I_n$). Proceed from there.