Hello,

$\displaystyle \mathcal{B}(\mathbb{R})$ denotes the Borel sigma-algebra over the real line.

For any $\displaystyle x\in\mathbb{R}$, let's denote $\displaystyle T_x$ the application $\displaystyle T_x(y)=x+y ~,~ y\in\mathbb{R}$

1. Show that $\displaystyle T_x(\mathcal{B}(\mathbb{R}))$ is a sigma-algebra over the real line.

2. Show that $\displaystyle T_x(\mathcal{B}(\mathbb{R}))$ contains all the intervals

3. Show that $\displaystyle \forall x\in\mathbb{R} ~,~ T_x(\mathcal{B}(\mathbb{R}))=\mathcal{B}(\mathbb{R })$

Okay, so the main problem is that I don't know how and where to start... my first idea was to start by what's asked in question 3., so I believe this wasn't the right idea lol...

Thanks for any help