Translational Invariance of a Sigma Algebra.

Question

There are several questions relating to this topic, but these tend to be about Borel $\sigma$-algebras or the generators of $\sigma$-algebras. I would instead like this question to be answered using the following definition;

Let $X$ be a set. A collection $\Sigma$ of subsets of $X$ is called a sigma-algebra if

1) $\emptyset \in \Sigma$;

2) $E \in \sigma$ implies $X\setminus E \in \Sigma $;

3) $E_n \in \Sigma, n\geq 1$ implies $\cup^{\infty}_{n=1}E_n \in \Sigma$.

My question is this:

Let $X = \mathbb{R}$ and let $x \in R$ be a fixed number. Prove that the collection of sets $\{A+x : A \in \Sigma \}$, denoted $\Sigma + x$, is a sigma-algebra of subsets of $\mathbb{R}$. Here $A+x=\{a+x: a\in A \}$.

The first criterion of the definition seems quite straightforward, but I am struggling with the other two.

Answer 1

Hint: use four things:

• $a+\varnothing=\varnothing$;

• $a+\bigcup_nA_n=\bigcup_n (a+A_n)$;

• $a+A^c=(a+A)^c$;

• by hypothesis, $\Sigma$ is a $\sigma$-algebra.