Borel sets in $\Bbb R (0,1]$ element of $B(\Bbb R)$

Question

Determine if interval $(0,1]$ is an element of $B(\Bbb R)$ Borel real numbers.

Can anyone help me with this? I know what sigma algebras are and Borel sets.

Also if $(0,1]$ is an element of $B(0,1]$ does that mean it is also element of $B(\Bbb R)$?

Answer 1

Note $(0,1]^c = (-\infty,0] \cup (1,\infty)$

$ (-\infty,0]$ is closed in $\mathbb R$ and $(1,\infty)$ is open in $\mathbb R$. Hence belongs borel sigma algebra.

Now by definition of sigma algebra, complement of the union of above sets belongs to sigma algebra. Hence interval $(0,1]$ is an element of borel sigma algebra on $\mathbb R$.