Measure Theory

First things

• Ring of sets
• Algebra of sets
• Sigma-ring
• Sigma-algebra

Measures

To start with we define rings, for example consider the ring of all half-open-half-closed rectangles of dimension $n$, call this $$\mathcal{J}^n$$

$$[[a,b))\in\mathcal{J}^n$$ means $$[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\in\mathcal{J}^n$$

This is clearly a ring, but not a $\sigma$-ring as for example $$\bigcup^\infty_{n=1}[[0,1-\tfrac{1}{n}))=[[0,1]]\notin\mathcal{J}^n$$