Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable/General Result
Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $n \in \N$.
Let $\sequence {X_i}_{i \mathop \in \N}$ be a sequence of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $\sequence {\alpha_i}_{i \mathop \in \N}$ be a sequence of real numbers.
Then:
- $\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is a real-valued random variable.
Proof
We proceed by induction.
For all $n \in \N$ let $\map P n$ be the proposition:
- $\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is $\Sigma$-measurable.
Basis for Induction
From the construction of $\sequence {X_i}_{i \mathop \in \N}$ we have:
- $X_1$ is $\Sigma$-measurable.
From Pointwise Scalar Multiple of Measurable Function is Measurable, we have:
- $\alpha_1 X_1$ is is $\Sigma$-measurable.
This is precisely $\map P 1$.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P n$ is true, where $n \ge 1$, then it logically follows that $\map P {n + 1}$ is true.
Our induction hypothesis is:
- $\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is $\Sigma$-measurable.
We aim to show that:
- $\ds \sum_{i \mathop = 1}^{n + 1} \alpha_i X_i$ is $\Sigma$-measurable.
Induction Step
This is our induction step.
We have:
- $\ds \sum_{i \mathop = 1}^{n + 1} \alpha_i X_i = \alpha_{n + 1} X_{n + 1} + \sum_{i \mathop = 1}^N \alpha_i X_i$
From our induction hypothesis, we have:
- $\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is $\Sigma$-measurable.
From the construction of $\sequence {X_i}_{i \mathop \in \N}$ we have:
- $X_{n + 1}$ is $\Sigma$-measurable.
So from Pointwise Scalar Multiple of Measurable Function is Measurable, we have:
- $\alpha_{n + 1} X_{n + 1}$ is $\Sigma$-measurable.
So, from Pointwise Sum of Measurable Functions is Measurable, we have:
- $\ds \alpha_{n + 1} X_{n + 1} + \sum_{i \mathop = 1}^n \alpha_i X_i = \sum_{i \mathop = 1}^{n + 1} \alpha_i X_i$ is $\Sigma$-measurable.
$\blacksquare$