Riemann algebra

Riemann algebra: Let $f,g:[a,b]\to\mathbb R$ be Riemann integrable on the interval $[a,b]$. Then the product $fg$ is Riemann integrable on $[a,b]$. Consequently, the collection of Riemann integrable functions on $[a,b]$ is closed under pointwise addition, scalar multiplication (linearity), and pointwise multiplication, hence forms an algebra over $\mathbb R$.