Almost All" Numbers in [0,1] are Satanic

[EL]

Could someone please help me with this problem which I have suffered many hours in front of? I never came to any solution which totally satisfied me, and I'm guessing I'm on the wrong track...

A number x in the interval [0,1] is called "satanic" if the decimal expansion of x contains somewhere the sequence 666.
Show that "almost all" numbers in [0,1] are satanic, i.e., that m([0,1]\S)=0 where S is the set of satanic numbers, and m is the Lebesgue measure.

 

[matt grime]

Let us assume [0,1]\S is measurable (I can't see why it shouldn't be), its measure is then the probability that a number I pick at random from [0,1] doesn't contain 666 at any point in its expansion. But that probability is zero, as the 666 occurs roughly once in every 1000 blocks of three digits, so the probablity it doesn't occur in the first n*1000 places is 999/1000)^n which tends to zero as n tends to infinity.

 

[M98Ranger]

First of all, I want to acknowledge the frustration and difficulty you have faced in trying to solve this problem. It can be disheartening when we spend hours trying to find a solution and still feel like we are on the wrong track. But don't give up, as it is often through perseverance and seeking help that we can eventually find a satisfying solution.

Now, let's address the issue at hand. The statement that "almost all" numbers in [0,1] are satanic can be proven using the concept of measure theory. In this context, the term "almost all" means that the set of numbers in [0,1] that are not satanic has a measure of 0. In other words, the set of satanic numbers is so small compared to the whole interval [0,1] that it can be considered negligible.

To prove this, we first need to define the Lebesgue measure, denoted by m, which is a mathematical tool used to measure the size of sets in a given space. In this case, the space is [0,1] and the set we are interested in is S, the set of satanic numbers. The Lebesgue measure of a set is defined as the length, area, or volume of the set, depending on the dimension of the space.

Now, we can use the definition of the Lebesgue measure to show that the set of satanic numbers has a measure of 0 in [0,1]. Since the decimal expansion of a number can be infinite, we can think of the numbers in [0,1] as points on a number line, with each point representing a unique decimal expansion.

Given that the decimal expansion of a satanic number contains the sequence 666 somewhere, we can conclude that the set of satanic numbers is countable. This means that we can list all the satanic numbers in a sequence, such as 0.666, 0.0666, 0.00666, and so on. Since the set is countable, it has a measure of 0, according to the definition of the Lebesgue measure.

On the other hand, the set of non-satanic numbers in [0,1] is uncountable, which means we cannot list all the numbers in a sequence. Therefore, the set of non-satanic numbers has a measure of 1, according to the Lebesgue measure. Since