Question

Suppose \(X_{1}\) and \(X_{2}\) are \(\mathscr{\text { -measurable random variables. Show that }}\) \(a_{1} X_{1}+a_{2} X_{2}\) is \(\mathscr{B}\)-measurable for all real \(a_{1}, a_{2}\).

Short answer

In conclusion, a linear combination \(a_1 X_1 + a_2 X_2\) of two \(\mathscr{B}\)-measurable random variables \(X_1\) and \(X_2\) is also \(\mathscr{B}\)-measurable, as we showed that the pre-image of the linear combination belongs to the sigma algebra \(\mathscr{B}\) for any Borel set and for all possible cases regarding \(a_1\) and \(a_2\).

Step-by-step solution

Define a Function for the Linear Combination

Let's define the function \(Y(\omega) = a_1 X_1(\omega) + a_2 X_2(\omega)\) representing the linear combination of \(X_1\) and \(X_2\).

Show that the Pre-Image of the Linear Combination Belongs to the Sigma Algebra

To prove that the linear combination is \(\mathscr{B}\)-measurable, we need to show that for any Borel set \(B \in \mathscr{B}\), the pre-image of the linear combination, \((Y^-1)(B)\), belongs to the sigma algebra generated by \(X_1\) and \(X_2\). Consider an arbitrary Borel set \(B \in \mathscr{B}\). By definition, for \(\mathscr{B}\)-measurable random variables \(X_1\) and \(X_2\), the pre-images \((X_1^-1)(B)\) and \((X_2^-1)(B)\) must belong to \(\mathscr{B}\). Now, let's examine the pre-image of the linear combination \((Y^-1)(B)\): $$Y^{-1}(B) = \{\omega \in \Omega : Y(\omega) \in B\} = \{\omega \in \Omega : a_1 X_1(\omega) + a_2 X_2(\omega) \in B\}.$$ We need to show that \((Y^-1)(B)\) belongs to \(\mathscr{B}\).

Consider the Case where \(a_1, a_2 \neq 0\)

If both \(a_1\) and \(a_2\) are non-zero, we can rewrite the pre-image as follows: $$Y^{-1}(B) = \{\omega \in \Omega : a_1 X_1(\omega) + a_2 X_2(\omega) \in B\} = \{\omega \in \Omega : X_1(\omega) \in \frac{1}{a_1}(B - a_2 X_2(\omega))\}.$$ Notice that \(\frac{1}{a_1}(B - a_2 X_2(\omega))\) is a Borel set, since it involves the difference and scaling of two Borel sets. Therefore, the pre-image of the linear combination, \((Y^-1)(B)\), can be written as the pre-image of the random variable \(X_1\) evaluated in a Borel set. As \(X_1\) is \(\mathscr{B}\)-measurable, the pre-image \((Y^-1)(B)\) must also belong to \(\mathscr{B}\).

Consider Degenerate Cases

Now let's consider the cases when \(a_1 = 0\) or \(a_2 = 0\) or both. 1. If \(a_1 = 0\) and \(a_2 \neq 0\), then \(Y(\omega) = a_2 X_2(\omega)\). Since \(X_2\) is \(\mathscr{B}\)-measurable, any scalar multiple of \(X_2\) is also \(\mathscr{B}\)-measurable. 2. Similarly, if \(a_2 = 0\) and \(a_1 \neq 0\), \(Y(\omega) = a_1 X_1(\omega)\), and \(Y\) is \(\mathscr{B}\)-measurable since \(X_1\) is \(\mathscr{B}\)-measurable. 3. If both \(a_1\) and \(a_2\) are \(0\), then \(Y(\omega) = 0\) is a constant function and hence \(\mathscr{B}\)-measurable. In each case, \(Y\) is \(\mathscr{B}\)-measurable for all real values of \(a_1\) and \(a_2\). In conclusion, \(a_1 X_1 + a_2 X_2\) is \(\mathscr{B}\)-measurable for all real \(a_1\) and \(a_2\), as we showed the pre-image of the linear combination belongs to the sigma algebra \(\mathscr{B}\) for any Borel set and for all possible cases regarding \(a_1\) and \(a_2\).