### A Mild Glimpse Into Measure Theory (2)

#### by tzy9393

**3. Measures and Outer measures**

Let be a measurable space.

**Definition:** A * measure* on is a function such that:

(1)

(2) For all families of disjoint measurable sets,

**Remarks:**

- is allowed
- Property (2) is called -additivity
- We can also derive

Proposition 6: Let , we have the following:

(1) If , then .

(2) If and , then

(3)

(4) If for all and for all , then .

(5) If for all and for all , then .

(6) If , then .

To be updated.

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