# What is the integral of the Cantor function?

## What is the integral of the Cantor function?

EDIT: More generally, if F is any continuous distribution function (the Cantor function f is a particular example), then ∫RF(x)dF(x)=1/2. As before, this can be proved using integration by parts, which is allowed since F is continuous and non-decreasing.

Is the Cantor function Riemann integrable?

Suppose f is the characteristic function of the Cantor set. So f(x)=1 when x is in the Cantor set, and f(x)=0 otherwise. This f surprisingly is Riemann integrable and its integral is 0.

What is the Devil’s Staircase math?

Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically rises from 0 to 1! Take any number X in the unit interval, and express it in base 3.

### Is Cantor function uniformly continuous?

Cantor function f is defined by So how to prove it is uniformly continuous and even how to get α=log2log3? it is enough to show that it is continuous since it is defined on a compact set. Look at the first base 3 digit where the two numbers differ.

Is Cantor function Lebesgue measurable?

We use the Cantor-Lebesgue Function to show there are measurable sets which are not Borel; so B ⊊ M.

Is Cantor set measurable?

In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure.

## Why is the Cantor set uncountable?

Perhaps the most interesting property is that it is also uncountable. In its construction we remove the same number of points as the number left behind to form the Cantor set, which leads us to this result. Theorem 2.1. The Cantor set is uncountable.

How do you prove 1/4 is in the Cantor set?

A more plodding way to show it is to look at the series 29+292+293+⋯=14. This shows that the base-3 expansion of 1/4 is 0.02020202…. Since it has a base-3 expansion with only 0s and 2s, it is in the Cantor set.

Why is Cantor set closed?

Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space. Since it is also totally bounded, the Heine–Borel theorem says that it must be compact.

### Is the Cantor function integral to the probability density function?

However, no non-constant part of the Cantor function can be represented as an integral of a probability density function; integrating any putative probability density function that is not almost everywhere zero over any interval will give positive probability to some interval to which this distribution assigns probability zero.

What is the Cantor function?

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous.

How do you find the Cantor distribution?

The Cantor function can also be seen as the cumulative probability distribution function of the 1/2-1/2 Bernoulli measure μ supported on the Cantor set: c ( x ) = μ ( [ 0 , x ] ) {\extstyle c(x)=\\mu ([0,x])} . This probability distribution, called the Cantor distribution, has no discrete part.

## How do you find the Cantor function for z = 1/3?

be the dyadic (binary) expansion of the real number 0 ≤ y ≤ 1 in terms of binary digits bk ∈ {0,1}. Then consider the function For z = 1/3, the inverse of the function x = 2 C1/3 ( y) is the Cantor function. That is, y = y ( x) is the Cantor function. In general, for any z < 1/2,…