We consider the family $$\mathcal{I}=\{(a,b)\;|\;-\infty<a<b<+\infty\}\cup\{[-\infty, b)\;|\;b\in\mathbb{R}\}\cup\{(a,+\infty]\;|\;a\in\mathbb{R}\}.$$
I proved that $$\mathcal{B}\big(\overline{\mathbb{R}}\big)=\sigma_0\big(\mathcal{I}\big),$$ where $\sigma_0\big(\mathcal{I}\big)$ is the $\sigma$-algebra generated by $\mathcal{I}.$
What I would like to prove. I would like to prove that $$\mathcal{B}\big(\overline{\mathbb{R}}\big)=\sigma_0\big(\mathcal{J}\big),$$ where $\mathcal{J}$ is the family $$\mathcal{J}=\big\{(a,+\infty]\;|\;a\in\mathbb{R}\big\}.$$
We observe that $\mathcal{J}\subseteq\mathcal{I}$, then $\sigma_0\big(\mathcal{J}\big)\subseteq\sigma_0\big(\mathcal{I}\big)=\mathcal{B}\big(\overline{\mathbb{R}}\big)$. Note that the inclusion $\sigma_0\big(\mathcal{J}\big)\subseteq\sigma_0\big(\mathcal{I}\big)$ is a conseguence of a proposition that I have already proved.
Question. How can I prove that $$\mathcal{B}\big(\overline{\mathbb{R}}\big)\subseteq\sigma_0\big(\mathcal{J}\big).$$
Thanks!