Let $\Omega=[0,1],\mathcal{J}=\{ \text{all intervals in }[0,1] \}$. For an interval $I\in \mathcal{J}$, set $\mathbb{P}(I)=\text{length of }I$. Given this setup, there is a σ-algebra $\mathcal{M}$ on $[0,1]$ with probability measure $\lambda$ such that $\mathcal{J}\subseteq \mathcal{M}$ and $\lambda(I)=\text{length of }I$.

There exists a probability space $(\Omega,\mathcal{M},\mathbb{P}^{*})$ such that $\Omega=[0,1],\mathcal{M}\supseteq\mathcal{J}$ and for any interval $I\subseteq[0,1]$, $\mathbb{P}^{*}(I)=\text{length of }I$. This triple is called the uniform distribution on $[0,1]$ or the Lebesgue measure on $[0,1]$.