Measure theory & Lebesgue integration

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Let $f = 3 \cdot \mathbf{1}_{[0, 2)} + 5 \cdot \mathbf{1}_{[2, 5)}$ be a simple function on $\mathbb{R}$ with Lebesgue measure $\lambda$. Compute the Lebesgue integral $\int_{\mathbb{R}} f \, d\lambda$.

2. Warm-up

   Let $f = 6 \cdot \mathbf{1}_{[0, 1)} + 4 \cdot \mathbf{1}_{[1, 2)}$ be a simple function on $\mathbb{R}$ with Lebesgue measure $\lambda$. Compute the Lebesgue integral $\int_{\mathbb{R}} f \, d\lambda$.

3. Warm-up

   Let $f = 5 \cdot \mathbf{1}_{[0, 2)} + 3 \cdot \mathbf{1}_{[2, 4)}$ be a simple function on $\mathbb{R}$ with Lebesgue measure $\lambda$. Compute the Lebesgue integral $\int_{\mathbb{R}} f \, d\lambda$.

Problems shown under their original terms: Generated (template, solver-computed).