Let's go through the solution step-by-step:

Problem Statement

The problem is to show that if a function $f$ is integrable, then it must be finite-valued almost everywhere (a.e.).

Solution Breakdown

1. Assumption: Assume for contradiction that $f$ is not finite-valued almost everywhere. This means that there exists a set $E \subset \mathbb{R}$ with positive measure $m(E) > 0$ where $f(x) = \infty$ for all $x \in E$.

2. Implication of $f$ being infinite on $E$: Since $f(x) = \infty$ on the set $E$, it implies that $|f(x)| = \infty$ on $E$. This leads us to consider the integral $\int |f| \, dx$ over the set $E$, which would theoretically be infinite because the function $|f|$ takes an infinite value on a set of positive measure.

3. Contradiction Setup: For any positive integer $n$, consider the following inequality: $\int_E |f| \, dx > n \cdot m(E)$ Since $|f| = \infty$ on $E$, this inequality will hold for any $n$ (no matter how large).

4. Integrability Condition: However, if $f$ is integrable over the entire space, $\int |f| \, dx$ should be finite. But, our assumption that $f = \infty$ on $E$ with $m(E) > 0$ implies that $\int_E |f| \, dx$ is infinite, which contradicts the assumption that $f$ is integrable.

5. Conclusion: Therefore, our assumption that $f$ takes infinite values on a set of positive measure must be false. Hence, $f$ is finite-valued almost everywhere, meaning that $f$ can only be infinite on a set of measure zero.

This concludes the proof.

Would you like further clarification on any of the steps?

Related Questions

1. What does it mean for a function to be integrable?
2. How does the concept of measure relate to integrability?
3. Can a function that takes on infinite values be integrable in certain contexts?
4. What does "finite-valued almost everywhere" mean in the context of integration?
5. Why is a contradiction a useful method in proofs like this one?

Tip:

When proving properties of integrable functions, it can be helpful to consider the behavior of the function over sets of positive measure and use inequalities to highlight contradictions with integrability assumptions.