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The statement provided in Danish translates to the following in English:

**(iii) Let \((X, \mathcal{A}, \mu)\) be a measure space. Let \(u \in \mathcal{M}_{\mathbb{R}^+}(\mathcal{A})\), and suppose that \(\int u \, d\mu < \infty\). Then \(u < \infty\) almost everywhere (i.e., \(\mu\)-almost everywhere).**

To assess whether this statement is true or false, let's break down the components and reasoning:

1. **Measure Space**: \((X, \mathcal{A}, \mu)\) is a measure space, which implies \(\mu\) is a measure defined on a sigma-algebra \(\mathcal{A}\) over a set \(X\).
   
2. **Function \(u\)**: \(u \in \mathcal{M}_{\mathbb{R}^+}(\mathcal{A})\) indicates that \(u\) is a measurable function taking values in the non-negative real numbers (i.e., \(u : X \to [0, \infty)\)).

3. **Integral Condition**: \(\int u \, d\mu < \infty\) means the integral of \(u\) with respect to \(\mu\) is finite, which implies that the function \(u\) is integrable over \(X\) with respect to the measure \(\mu\).

4. **Conclusion**: The statement asserts that under these conditions, \(u < \infty\) holds almost everywhere (a.e.) with respect to \(\mu\).

### Reasoning
If \(u\) were infinite on a set of positive measure, then the integral \(\int u \, d\mu\) would be infinite, contradicting the given condition that \(\int u \, d\mu < \infty\). Thus, for the integral to be finite, \(u\) must be finite on all but a \(\mu\)-null set. This is the basis for concluding that \(u < \infty\) almost everywhere.

### Conclusion
The statement is **true**. 

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. What does it mean for a function to be integrable on a measure space?
2. How is almost everywhere defined in terms of measure theory?
3. What are the implications if \(\int u \, d\mu\) diverges instead?
4. Can \(u\) be bounded and still satisfy \(\int u \, d\mu < \infty\)?
5. What role does the choice of measure \(\mu\) play in the finiteness of \(\int u \, d\mu\)?

**Tip:** In measure theory, the concept of "almost everywhere" allows for generalizations that bypass sets of measure zero, making it a powerful tool for analyzing functions with infinite or undefined values on small sets.

Let \((X, \mathcal{A}, \mu)\) be a measure space. Let \(u \in \mathcal{M}_{\mathbb{R}^+}(\mathcal{A})\), and suppose that \(\int u \, d\mu < \infty\). Then \(u < \infty\) almost everywhere.

Explore the concept of integrability in measure theory and the condition for a function to be finite almost everywhere. Given a measure space \((X, \mathcal{A}, \mu)\) and a function \(u \in \mathcal{M}_{\mathbb{R}^+}(\mathcal{A})\) with finite integral, understand why \(u\) must be finite almost everywhere.

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