To prove that if $f$ is integrable, then $f$ is finite-valued almost everywhere (a.e.), we need to use the fact that integrability implies that the function has finite integral over its domain, and this, in turn, implies that the function cannot take infinite values on a set of positive measure.

Proof:

Let $f$ be a measurable function on a measure space $(X, \mathcal{M}, \mu)$, where $\mu$ is the measure, and assume that $f$ is integrable, i.e.,

$\int_X |f(x)| \, d\mu(x) < \infty.$

This means the absolute value of $f$ has a finite integral over the set $X$.

We want to show that $f$ is finite-valued almost everywhere (a.e.), meaning that the set where $f$ is infinite must have measure zero, i.e.,

$\mu(\{ x \in X \mid |f(x)| = \infty \}) = 0.$

Step 1: Decompose the domain

We decompose the domain $X$ into two sets:

1. $A = \{ x \in X \mid |f(x)| = \infty \}$, the set where $f$ takes infinite values.
2. $B = \{ x \in X \mid |f(x)| < \infty \}$, the set where $f$ is finite.

Since $f$ is integrable, the integral of $|f|$ over the entire domain is finite:

$\int_X |f(x)| \, d\mu(x) = \int_A |f(x)| \, d\mu(x) + \int_B |f(x)| \, d\mu(x).$

Step 2: Behavior on set $A$

On the set $A$, $|f(x)| = \infty$. If $\mu(A) > 0$, then the integral over $A$ would be infinite because $|f(x)| = \infty$ for all $x \in A$. Specifically:

$\int_A |f(x)| \, d\mu(x) = \infty \cdot \mu(A).$

For the integral of $|f(x)|$ to remain finite, we must have $\mu(A) = 0$. Otherwise, the integral would be infinite, contradicting the assumption that $f$ is integrable.

Step 3: Conclusion

Since $\mu(A) = 0$, we conclude that $f(x)$ is finite almost everywhere. More precisely:

$\mu(\{ x \in X \mid |f(x)| = \infty \}) = 0.$

Thus, $f$ is finite-valued almost everywhere.

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Would you like further details or have any questions? Here are some related questions to expand on this:

1. What does it mean for a function to be integrable in Lebesgue sense?
2. Can a function be unbounded but still integrable?
3. How does the proof change if we work on a specific measure, like the Lebesgue measure?
4. Is it possible for a non-integrable function to be finite almost everywhere?
5. How does the concept of "almost everywhere" apply in real-world contexts?

Tip: Remember that integrable functions have more restrictions on their behavior compared to general measurable functions, ensuring that they can't have "too large" values on sets of positive measure.