This example from your book discusses a measurable function $f$ and an integrable function $g$, along with real constants $\alpha$ and $\beta$. It addresses the existence of a number $\gamma$ that lies between $\alpha$ and $\beta$ and satisfies the equation:

$\int f |g| \, dx = \gamma \int |g| \, dx$

Explanation of the Example:

Given:

• $f$ is measurable.
• $g$ is integrable.
• $\alpha \leq f \leq \beta$ almost everywhere (a.e.).

The goal is to show that there exists some $\gamma$ between $\alpha$ and $\beta$ such that the integral of $f \cdot |g|$ equals $\gamma$ times the integral of $|g|$.

Solution:

1. Bounding $f \cdot |g|$: Since $\alpha \leq f \leq \beta$ a.e., multiplying by $|g|$ gives:

   $\alpha |g| \leq f |g| \leq \beta |g| \quad \text{a.e.}$

2. Integrating both sides: Since $g$ is integrable, you can integrate the inequality over the entire domain:

   $\alpha \int |g| \, dx \leq \int f |g| \, dx \leq \beta \int |g| \, dx$

   This shows that $\int f |g| \, dx$ lies between $\alpha \int |g| \, dx$ and $\beta \int |g| \, dx$.

3. Conclusion: Since the integral of $f |g|$ is bounded between these two integrals, by the Intermediate Value Theorem (for integrals), there exists some $\gamma \in [\alpha, \beta]$ such that:

   $\int f |g| \, dx = \gamma \int |g| \, dx$

Special Case Considerations:

• If $\int |g| \, dx = 0$: The function $g$ must be 0 almost everywhere (a.e.), and in this case, the result is trivial because both sides of the equation are 0.

• If $\int |g| \, dx \neq 0$: You can solve for $\gamma$ as:

  $\gamma = \frac{\int f |g| \, dx}{\int |g| \, dx}$

  This result directly follows from the division of both sides by $\int |g| \, dx$, assuming it is non-zero.

Let me know if you'd like further clarification on any part of the example.

Here are 5 related questions that expand on this concept:

1. What are the conditions for the Intermediate Value Theorem to apply to integrals?
2. How does integrability of $f$ and $g$ influence the result?
3. Can the value $\gamma$ ever lie outside $[\alpha, \beta]$?
4. What changes if $g$ is not integrable but measurable?
5. How would this example change if $f$ is not measurable?

Tip: Always check whether the integral bounds are zero, as this could simplify the problem significantly by making certain parts trivial.