Prove that the function is monotonic and not decreasing

In summary, the exercise deals with measure theory and Borel sets. To prove the given statements, the definition of $f_{\mu}$ and the properties of measures are used.
  • #1
fabiancillo
27
1
Hello, I don't know to solve this exercise:

Let $\mathcal{B}_\mathbb{R}$ the $\sigma-algebra$ Borel in $\mathbb{R}$ and let $\mu : \mathcal{B}_\mathbb{R} \rightarrow{} \mathbb{R}_{+}$ a finite measure. For each $x \in \mathbb{R}$ define

$$f_{\mu} := \mu((- \infty,x]) $$

Prove that:

a) $f_{\mu}$ is a monotonic non-decreasing function
b) $\mu((a,b]) = f_{\mu}(b)- f_{\mu}(a)$ for all $a,b \in \mathbb{R}$

The definition ($\sigma-algebra$ borel , is this:

Definition $\sigma-algebra$ Borel : Let $(X,\tau)$ a topological space. we define the borelian tribe associated with $(X, \tau)$ as the algebra generated by T, that is

$$\mathcal{B}(X)= \mathcal{B}(X,\tau)= \sigma (\tau)$$

I need a hint.Thanks
 
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  • #2
Hint: For part (a), use the fact that $\mu$ is a measure and that $(-\infty,x]$ is an interval with endpoints $-\infty$ and $x$.For part (b), use the definition of the function $f_\mu$ and the fact that $\mu$ is a measure.
 
  • #3
for reaching out for help on this exercise! It looks like you're working with some measure theory and Borel sets. To prove that $f_{\mu}$ is a monotonic non-decreasing function, you can start by considering two points $x_1, x_2 \in \mathbb{R}$ such that $x_1 < x_2$. Then, think about the definition of $f_{\mu}$ and how it relates to the measure $\mu$. Can you use this to show that $f_{\mu}(x_1) \leq f_{\mu}(x_2)$?

For the second part of the exercise, you can use the definition of $f_{\mu}$ again to show that $f_{\mu}(b) - f_{\mu}(a) = \mu((- \infty, b]) - \mu((- \infty, a])$. Then, use the properties of measures to see how this equals $\mu((a, b])$.

Hope this helps! Good luck with the rest of the exercise.
 

1. What does it mean for a function to be monotonic?

A monotonic function is one that either always increases or always decreases as its input increases. In other words, the function's output either always increases or always decreases as its input increases.

2. How can I prove that a function is monotonic?

To prove that a function is monotonic, you must show that the function either always increases or always decreases. This can be done by taking the derivative of the function and showing that it is always positive (increasing) or always negative (decreasing).

3. What is the difference between a monotonic and a non-monotonic function?

A monotonic function always increases or always decreases, while a non-monotonic function may increase and decrease at different points. In other words, a non-monotonic function does not have a consistent trend of increasing or decreasing.

4. Can a function be both monotonic and not decreasing?

Yes, a function can be both monotonic and not decreasing. This means that the function is always increasing or staying the same, but it is not necessarily always increasing.

5. Why is it important to prove that a function is monotonic and not decreasing?

Proving that a function is monotonic and not decreasing is important because it allows us to make conclusions about the behavior of the function. For example, we can use this information to determine the maximum or minimum values of the function, or to make predictions about its future behavior.

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