E Measurable Implies E + y Measurable

e(ho0n3

The problem statement, all variables and given/known data
Show that if a subset E or R is measurable, then each translate E + y of E is also measurable.

Relevant equations
E is measurable if for all subsets A of R

$$m^*(A) = m^*(A \cap E) + m^*(A \cap (\textbf R \setminus E))$$

where m* is the outer measure.

The attempt at a solution
The result is trivial if y = 0, so supposes y is not 0. I will denote the complement of a subset of R using '. Let F = E + y. By monotonicity of m*,

$$m^*(A) \le m^*(A \cap F) + m^*(A \cap F')$$

In the other direction, I have determined the following: Since E is a subset of F' and F is a subset of E', we have that $A \cap E \subseteq A \cap F'$ and $A \cap F \subseteq A \cap E'$. By monotonicity of m*, $m^*(A \cap E) \le m^*(A \cap F')$ and $m^*(A \cap F) \le m^*(A \cap E')$. The former inequality is no good, but the latter one isn't. I don't know how to proceed from here. Any tips?

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving