Measure theory and integration

Let $\mathcal A$ be the collection of subsets $E \subseteq \N$ such that $E$ is finite or $\N \setminus E$ is finite. Which statement is true?

Let $f = 5 \cdot \mathbf 1_{\Q \cap [0, 1]} + 3 \cdot \mathbf 1_{[0, 2]}$ on $\R$. Compute the Lebesgue integral $\int f \, d\lambda$.

Let $g_n = \mathbf 1_{[n, n+1]}$ on $\R$ with Lebesgue measure. Then $g_n \to 0$ pointwise, but $\int g_n = 1$ for all $n$. Which hypothesis of the dominated convergence theorem fails?

The proof of Fatou's lemma sets $g_k = \inf_{n \ge k} f_n$ and concludes $\int \liminf f_n = \lim_k \int g_k$. Which result justifies this equality?

For $m, n \ge 0$ let $a_{m,n} = 1$ if $m = n$, $a_{m,n} = -1$ if $m = n + 1$, and $a_{m,n} = 0$ otherwise. Compute $\displaystyle \sum_{m=0}^\infty \Big( \sum_{n=0}^\infty a_{m,n} \Big)$.

Which Lean statement correctly formalises Markov's inequality $\varepsilon\,\mu(\{f \ge \varepsilon\}) \le \int f \, d\mu$ for measurable $f : \alpha \to [0, \infty]$? (Here ℝ≥0∞ is $[0, \infty]$ and ∫⁻ is the integral of such functions.)

Select all true statements.

In the proof that $L^p$ is complete, given a Cauchy sequence $(f_n)$ we choose $n_1 < n_2 < \cdots$ with $\|f_{n_{j+1}} - f_{n_j}\|_p < 2^{-j}$. Why is this subsequence needed?

Write union_length(intervals) that returns the Lebesgue measure of a finite union of closed intervals, given as a list of pairs (a, b) with a <= b. The intervals may overlap. (Countable subadditivity says the answer is at most the sum of the lengths; overlaps make it smaller.)