Why couldn't monotonicity be used again in examples 2 and 4 of real analysis?

  • Thread starter nateHI
  • Start date
In summary, the conversation is discussing the use of monotonicity in two examples from a text on exterior measures. While example 2 uses monotonicity to conclude that the outer measure of a cube is less than or equal to its volume, example 4 requires a more complicated argument for a rectangle. This is because the exterior measure is defined using coverings by closed cubes, which explains the difference in the two examples. The inequalities in question are not related to monotonicity.
  • #1
nateHI
146
4
**Moderator: Please move my thread to the correct section. Sorry about posting it in the wrong location**

http://press.princeton.edu/chapters/s8008.pdf

In example 2 on page 11 of the text I linked above, the book uses the monotonicity property of the outer measure to conclude
##m_*(Q) \le |Q|## where Q is a cube.
However, in example 4 on the next page they require a more complicated argument to reach the conclusion
##m_*(R) \le |R|## where R is a rectangle.
My question is, why couldn't they use monotonicity again?
 
Last edited:
Physics news on Phys.org
  • #2
The text is confusing!
 
  • #3
They define the exterior measure using coverings by closed cubes. Example 2 is an closed cube, and Example 4 is a rectangle, and that's what accounts for the difference. In fact, monotonicity has nothing to do with the inequalities you are worried about.
 

1. What is monotonicity in real analysis?

Monotonicity in real analysis refers to the behavior of a function in terms of its increasing or decreasing nature. A function is considered monotonic if it maintains a consistent trend of either increasing or decreasing values over its entire domain.

2. How is monotonicity different from continuity?

While continuity refers to the smoothness of a function without any sudden jumps or breaks, monotonicity focuses on the direction of change in a function. A continuous function can still be non-monotonic if it has areas of both increasing and decreasing values, while a monotonic function must maintain a consistent trend.

3. What is the relationship between monotonicity and differentiability?

A differentiable function must also be continuous, but it does not necessarily have to be monotonic. However, if a function is monotonic and has a derivative, then the derivative must be either always positive or always negative, depending on the direction of monotonicity.

4. How is monotonicity related to convexity and concavity?

In real analysis, a function is considered convex if its graph curves upwards, while it is concave if its graph curves downwards. A convex (or concave) function must also be monotonic, as it maintains a consistent trend of either increasing or decreasing values.

5. Can a function be both monotonic and oscillating?

No, a function cannot be both monotonic and oscillating. Monotonicity requires a consistent trend of either increasing or decreasing values, while oscillation refers to the behavior of a function that alternates between increasing and decreasing values. A function can be either monotonic or oscillating, but not both.

Similar threads

I Using reference frames for derivation of Lagrangian
    • Mechanics
Replies
25
Views
1K
MHB What Can You Create with TikZ Code Examples?
    • MATLAB, Maple, Mathematica, LaTeX
Replies
3
Views
4K
I Some questions in QFT (EM vertex, Ward identity, etc.)
    • Quantum Physics
Replies
5
Views
2K
B Calc Excess Rad on Earth in Curved 3D Space | Feynman Lectures 6-2
    • Special and General Relativity
Replies
29
Views
1K
A Ballentine on the "multicomponent state function"
    • Quantum Physics
Replies
1
Views
850
MHB Splitting Fields - Dummit and Foote - Example - page 541 - x^p - 2, p prime
    • Linear and Abstract Algebra
Replies
4
Views
2K
MHB Sensitivity Analysis model for omnivory systems (coupled system of ode's)
    • MATLAB, Maple, Mathematica, LaTeX
Replies
0
Views
1K
Challenge Math Challenge Thread (October 2023)
    • Math Proof Training and Practice
Replies
25
Views
2K
Challenge Math Challenge - June 2020
    • Math Proof Training and Practice
5
Replies
156
Views
15K
I P & Q branches confusion in rovibrational transitions + rant
    • Atomic and Condensed Matter
Replies
7
Views
3K

Hot Threads

Recent Insights

Back
Top