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

In summary, to approach the problem, start with an arbitrary ##\epsilon \gt 0## and use the definition of the (outer) measure of a set to find a countable set of open intervals, ##C_{epsilon}##, covering ##E\alpha## with a summed length less than ##m( E\alpha) + \epsilon##. Then, use the definition of the derivative to find smaller intervals, ##I_n##, within each interval of ##C_{epsilon}## whose image under ##f## is of length smaller than ##\alpha## length(##I_n##). From there, continue with the problem.
  • #1
laurabon
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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 ??}
 
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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.
 
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1. What is the definition of "measure" in the context of a set?

The measure of a set is a numerical representation of its size or extent. It is a way to quantify the elements or elements' properties within the set.

2. How is the measure of a set determined?

The measure of a set is determined by applying a specific mathematical function or formula to the elements in the set. The function used may depend on the type of set and the properties being measured.

3. What are some common measures used for sets?

Some common measures used for sets include cardinality, which counts the number of elements in a set, and probability, which measures the likelihood of an event occurring within a set. Other measures may include length, area, volume, and density.

4. How is the measure of a set affected by adding or removing elements?

The measure of a set can be affected by adding or removing elements. For example, if elements are added to a set, the measure will increase, and if elements are removed, the measure will decrease. However, the extent of the change in measure will depend on the specific measure being used and the properties of the added or removed elements.

5. Can the measure of a set be negative?

In some cases, the measure of a set can be negative. This may occur when the elements in the set have properties that result in a negative value when the measure is calculated. However, negative measures are not always meaningful or relevant, so it is important to consider the context and interpretation of the measure when dealing with negative values.

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