Finite Union of Sets in Subadditive Function

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Theorem

Let $\mathcal A$ be an algebra of sets.

Let $f: \mathcal A \to \overline {\R}$ be a subadditive function.

Let $A_1, A_2, \ldots, A_n$ be any finite collection of elements of $\mathcal A$.

Then:

$\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n f \left({A_i}\right)$


That is, for any finite collection of elements of $\mathcal A$, $f$ of their union is less than or equal to the sum of $f$ of the individual elements.


Proof

Proof by induction:

In the below, we assume that $A_1, A_2, \ldots$ are all elements of $\mathcal A$.


For all $n \in \N_{>0}$, let $P \left({n}\right)$ be the proposition:

$\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n f \left({A_i}\right)$


$P(1)$ is trivially true, as this just says $f \left({A_1}\right) \le f \left({A_1}\right)$.


Basis for the Induction

$P(2)$ is the case $f \left({A_1 \cup A_2}\right) \le f \left({A_1}\right) + f \left({A_2}\right)$, which comes from the definition of a subadditive function.

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:

$\displaystyle f \left({\bigcup_{i \mathop = 1}^k A_i}\right) \le \sum_{i \mathop = 1}^k f \left({A_i}\right)$

Then we need to show:

$\displaystyle f \left({\bigcup_{i \mathop = 1}^{k+1} A_i}\right) \le \sum_{i \mathop = 1}^{k+1} f \left({A_i}\right)$


Induction Step

This is our induction step:

\(\displaystyle f \left({\bigcup_{i \mathop = 1}^{k+1} A_i}\right)\) \(=\) \(\displaystyle f \left({\bigcup_{i \mathop = 1}^k A_i \cup A_{k+1} }\right)\)
\(\displaystyle \) \(\le\) \(\displaystyle f \left({\bigcup_{i \mathop = 1}^k A_i}\right) + f \left({A_{k+1} }\right)\) from the base case
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^k f \left({A_i}\right) + f \left({A_{k+1} }\right)\) from the induction hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^{k+1} f \left({A_i}\right)\)


So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\displaystyle \forall n \in \N_{>0}: \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n f \left({A_i}\right)$

$\blacksquare$