Sums, Multiples, and Products of Measurable Functions

Recall from the General Measurable Functions page that if $(X, \mathcal A)$ is a measurable space then an extended real-valued function $f$ defined on a measurable set $E$ is said to be measurable function on $E$ if for all $\alpha \in \mathbb{R}$ the following sets are measurable:

(1)

\begin{align} \quad \{ x \in E : f(x) < \alpha \} \end{align}

We will now state some important results regarding the linearity of measurable functions. These can be proven similarly to the analogous results proven for Lebesgue measurable functions.

Theorem 1 (Linearity of Measurable Functions): Let $(X, \mathcal A)$ be a measurable space and let $f$ and $g$ be measurable functions defined on a measurable set $E$, and let $c \in \mathbb{R}$. Then: a) $f + g$ is a measurable function on $E$. b) $cf$ is a measurable function on $E$.

Similarly, we can state a result for products of measurable functions.

Theorem 2: Let $(X, \mathcal A)$ be a measurable space and let $f$ and $g$ be measurable functions defined on a measurable set $E$. Then $fg$ is a measurable function on $E$.