### MA2224: Lebesgue integral

## Solutions to tutorial sheets

Solutions to tutorial sheets should appear here soon after the tutorial.

Links should open PDF format files.

- Tutorial Sheet 1 [February 5]
- Inverse and forward images. Example of an algebra.
- Tutorial Sheet 2 [February 9]
- Example on the interval algebra. Additivity, countable additivity and subadditivity.
- Tutorial Sheet 3 [February 19]
- Examples with the fact that $\mathscr L$ is a $\sigma$-algebra. Countable sets and the Cantor set included.
- Tutorial Sheet 4 [March 5]
- Measures of increasing unions. Measurability of limits of increasing sequences of functions. Example of a non-mesaurable function.
- Tutorial Sheet 5 [March 12]
- Basic examples of measurable functions. Compositions of continuos functions applied to measurable functions are measurable. Case of measurable functions defined on $[0,1]$/
- Tutorial/excercise Sheet 6 [due March 21]
- Standard forms for simple functions and integrals of nonnegative simple functions.
- Tutorial Sheet 7 [March 26]
- Monotone convergence theorem examples.
- Tutorial/excercise Sheet 8 [due April 13]
- Computing $\int_{\mathbb R} f \, d\mu$ via the Riemann integral (over finite closed intervals) and limits.