Linear Combination of Measures

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $\mu, \nu$ be measures on $\left({X, \Sigma}\right)$.


Then for all positive real numbers $a, b \in \R_{\ge 0}$, the pointwise sum:

$a \mu + b \nu: \Sigma \to \overline \R, \ \left({a \mu + b \nu}\right) \left({E}\right) := a \mu \left({E}\right) + b \nu \left({E}\right)$

is also a measure on $\left({X, \Sigma}\right)$.


Proof

Verifying the axioms $(1)$, $(2)$ and $(3')$ for a measure in turn:


Axiom $(1)$

The statement of axiom $(1)$ for $a \mu + b \nu$ is:

$\forall E \in \Sigma: \left({a \mu + b \nu}\right) \left({E}\right) \ge 0$


Let $E \in \Sigma$.

Then $\mu \left({E}\right), \nu \left({E}\right) \ge 0$ as $\mu$ and $\nu$ are measures.

Hence, $a \mu \left({E}\right) \ge 0$ as $a \ge 0$.

Also, $b \nu \left({E}\right) \ge 0$ since $b \ge 0$.


Therefore it follows that:

$a \mu \left({E}\right) + b \nu \left({E}\right) \ge 0$

as desired.

$\Box$


Axiom $(2)$

Let $\left({E_n}\right)_{n \in \N}$ be a sequence of pairwise disjoint sets in $\Sigma$.


The statement of axiom $(2)$ for $a \mu + b \nu$ is:

$\displaystyle \left({a \mu + b \nu}\right) \left({\bigcup_{n \mathop \in \N} E_n}\right) = \sum_{n \mathop \in \N} \left({a \mu + b \nu}\right) \left({E_n}\right)$


So let us do a direct computation:

\(\displaystyle \left({a \mu + b \nu}\right) \left({\bigcup_{n \mathop \in \N} E_n}\right)\) \(=\) \(\displaystyle a \mu \left({\bigcup_{n \mathop \in \N} E_n}\right) + b \nu \left({\bigcup_{n \mathop \in \N} E_n}\right)\) Definition of Pointwise Addition
\(\displaystyle \) \(=\) \(\displaystyle a \sum_{n \mathop \in \N} \mu \left({E_n}\right) + b \sum_{n \mathop \in \N} \nu \left({E_n}\right)\) $\mu$ and $\nu$ are measures and satisfy $(2)$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop \in \N} a \mu \left({E_n}\right) + b \nu \left({E_n}\right)\) Combined Sum Rule for Real Sequences
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop \in \N} \left({a \mu + b \nu}\right) \left({E_n}\right)\)

which establishes $a \mu + b \nu$ satisfies $(2)$.

$\Box$

Axiom $(3')$

The statement of axiom $(3')$ for $a \mu + b \nu$ is:

$\left({a \mu + b \nu}\right) \left({\varnothing}\right) = 0$


This is verified by the following:

\(\displaystyle \left({a \mu + b \nu}\right) \left({\varnothing}\right)\) \(=\) \(\displaystyle a \mu \left({\varnothing}\right) + b \nu \left({\varnothing}\right)\) Definition of Pointwise Addition
\(\displaystyle \) \(=\) \(\displaystyle a \cdot 0 + b \cdot 0\) $\mu$ and $\nu$ are measures and satisfy $(3')$
\(\displaystyle \) \(=\) \(\displaystyle 0\)

Thus, $a \mu + b \nu$ satisfies $(3')$.

$\Box$


Having verified an appropriate set of axioms, it follows that $a \mu + b \nu$ is a measure.

$\blacksquare$


Sources