What are measurable sets?

Question

1. Let $(X \times R,m\times m', \mu \times \lambda)$ be a measure space. If $E$ is any measurable set in $X \times R$ and if $\alpha,\beta \in R$ s.t $\alpha>0$, then $\{(x,y)~:~ (x,\alpha y+\beta)\in E\}$ is a measurable subset of $X\times R$.

2. Let $m$ be the Lebesgue measure on $[0,1]$ and let $\lambda$ be the counting measure on $\mathbb{N}$. Find all $(m\times\lambda)$-measurable sets.

3. Define $\mu(E)=\int_E ie^{ix} \, dm(x)$ for all Borel sets $E\subset [0,2\pi]$, where $m$ is the Lebesgue measure. Find all $\mu$-measurable sets.

Answer 1

Comment too long for comment box.

Unfortunately in measure theory, we use the word "measurable" to describe a variety of different concepts. For sets, there are three most commonly used situations where we call a set measurable.

1. If $(X, \mathscr{A})$ is a measurable space, then every $A \in \mathscr{A}$ is called measurable. Sometimes, if $(X ,\mathscr{A}, \mu)$ is a measure space and $\mathscr{A}$ is clear from the context of knowing what $\mu$ is, then each $A \in \mathscr{A}$ is sometimes referred to as $\mu$-measurable.

2. If $\mu^*$ is an outer measure on $X$, then the elements of $$\mathscr{M}_{\mu^*}:=\{E \subseteq X : \forall A \subseteq X, \mu^*(A) = \mu^*(A\cap E) + \mu^*(A \cap E^c) \}$$ are called $\mu^*$-measurable, or $\mu$-measurable if $\mu = \mu^*$ restricted to $\mathscr{M}_{\mu^*}$. (These are those sets measurable in the sense of (1.) on the measure space $(X,\mathscr{M}_{\mu^*},\mu)$). There is an equivalent formulation of this one in terms of outer and inner measures being equal.

3. If $(X, \mathscr{A}, \mu)$ is a measure space and $$\mathscr{A}_{\mu}:=\{A \subseteq X : \exists E,F \in \mathscr{A}, E \subseteq A \subseteq F, \mu(F\setminus E) = 0\},$$ then elements of $\mathscr{A}_\mu$ are called $\mu$-measurable. (These are those sets measurable in the sense of (1.) for the completion $\overline{\mu}$ of $\mu$, on the completed measure space $(X,\mathscr{A}_\mu,\overline{\mu})$).

For clarity and because it is most general, I always stick to (1.) and specify the $\sigma$-algebra if there is any chance of confusion.

Try seeing if any of these matches up with the definition in the book where you got those problems from.