📂Functions

Additive and Multiplicative Functions

Given a function $f : X \to Y$, let $a, b \in X$, $a_{i} \in X\ (i=1,\cdots)$.

Subadditive Function

A function $f$ is called a subadditive function when it satisfies the following equation:

$$f(a+b) \le f(a)+f(b)$$

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The absolute value is an example.

$$|3+(-4)| \le |3|+|-4|$$

Another example, if we have $f(x)=2x+3$ then

$$13=f(2+3) \le f(2)+f(3)=7+9=16$$

Additive Function

A function $f$ is called an additive function when it satisfies the following equation:

$$f(a+b) = f(a)+f(b)$$

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This is the case where equality holds in subadditivity.

For example, if $f(x)=4x$

$$20=f(2+3)=f(2)+f(3)=20$$

If set $E_{1},\ E_2$ satisfies $E_{1} \cap E_2 = \emptyset$ and let the number of elements in $n(E_{i})=E_{i}$ be

$$n(E_{1} \cup E_2) = n(E_{1}) + n(E_2)$$

Countably Subadditive Function

A function $f$ is called a countably subadditive function when it satisfies the following equation:

$$f \left( \sum_{i=1}^\infty a_{i} \right) \le \sum \limits_{i=1}^\infty f(a_{i})$$

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From subadditivity and additivity, it can be seen that this also holds for any arbitrary number of $N$ elements. If it holds for a countable number of elements, it is said to have countable subadditivity. An example of countable subadditivity is outer measure.

Countably Additive Function

A function $f$ is called a countably additive function when it satisfies the following equation:

$$f \left( \sum_{i=1}^\infty a_{i} \right) = \sum \limits_{i=1}^\infty f(a_{i})$$

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This is the case where equality holds in countable subadditivity.

For distinct elements, the outer measure is countably additive. If $E_{i} \cap E_{j} =\emptyset \quad \forall\ i,j$ then

$$\mu^{\ast} \left( \bigsqcup _{i=1}^\infty E_{i} \right) = \sum _{i=1}^\infty \mu^{\ast}(E_{i})$$

Submultiplicative Function

A function $f$ is called a submultiplicative function when it satisfies the following equation:

$$f(ab) \le f(a)f(b)$$

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This applies the same properties of addition to multiplication.

Multiplicative Function

A function $f$ is called a multiplicative function when it satisfies the following equation:

$$f(ab) = f(a)f(b)$$

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This is the case where equality holds in submultiplicativity.