Measure Lesbegue is a new topic for me and I don't know how I can start do this prove.
My request is:
Prove measurability in terms of a set of Lesbegue $A:=\bigcup_{n=1}^\infty [n,n+\frac{1}{8^n}]$.
What steps should I perform. Please instructions on what to do with this type of evidence.
OK, you are really new. First thing you should know is what is a sigma-algebra.
So, let $X$ be a nonempty set and let $P(X)$ be its partitive set. Now, a subset $M$ of $P(X)$ satisfying: 1. $X \in M$ 2. $(\forall n \in \mathbb{N}) A_n \in M \Longrightarrow \bigcup_{n\in \mathbb{N}} A_n \in M$\ 3. $A \in M \Longrightarrow A^c \in M$, where complement is taken over M.
The first thing about sigma-algebras that you should know is that the intersection of two sigma-algebras is again a sigma-algebras, leading to intersection of any family of sigma-algebras that do contain a certain set is again a sigma-algebra containing that set, we call such a sigma-algebra a sigma-algebra generated by that set.
Now, if you are in dimension n, note by $I_n$ the set of all $[a_1,b_1] \times [a_2,b_2] \times.... \times [a_n, b_n]$, the sigma-algebra generated by such a set is n-dimensional Lebesgue sigma-algebra and for those sets we say they are Lebesque measurable.
Now, you have union of segments, each of which is in $I_1$, so their union is in the sigma-algrabra generated by $I_1$ so it is Lebesgue measurable.