countable additivity

Countable Additivity

If $(A_i{)}_{i=1}^{\infty }$ is a countable sequence of disjoint sets, then the measure of the union of the sets is equal to the sum of the measures of the individual sets:

$$\mu \left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }\mu (A_i)$$

When you assemble a whole from non-overlapping countably many parts, “how much stuff” in the whole is exactly the sum of “how much” in each part – even when you keep adding parts forever.

Countability lets us list the pieces $A_1,A_2,\dots$ so we can form a series of unions:

$$U_n=\bigcup _{i=1}^n{A}_i$$

and if we measure each $U_n$ (which makess it nonnegative and take the limit) we get:

$$\mu \left(\bigcup _{i=1}^{\infty }{A}_i\right)=\underset{n\to \infty }{\lim }\mu (U_n)=\underset{n\to \infty }{\lim }\sum _{i=1}^n\mu (A_i)=\sum _{i=1}^{\infty }\mu (A_i)$$

I.e. we get a series which converges to the measure of the whole.

Disjoint, to avoid double-counting, which we’d need to remove via inclusion-exclusion principle.

Taking the cardinality as the measure

Additivity of cardinality

For two disjoint sets $A,B$, the cardinality of their union is the sum of the cardinalities.

$$A\cap B=\varnothing ⟹|A\cup B|=|A|+|B|$$

For nondisjoint sets, you need the inclusion-exclusion principle to remove the double counting of the intersection.

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