# What are measurable sets?

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.

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If you are confused about what measurable sets are, I suggest you focus on that before asking more advanced questions like the ones in the body of your post. –  Zev Chonoles Aug 18 '13 at 13:54
Additionally: to get the best possible answers, you should explain what your thoughts on the problem are. That way, people won't tell you stuff you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. –  Zev Chonoles Aug 18 '13 at 13:55
unfortunately I can't solve this kind of problems exactly .My teacher wants to certain answer! –  mahdieh Aug 19 '13 at 4:44

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.

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unfortunately I can't solve this kind of problems exactly .My teacher wants to certain answer! –  mahdieh Aug 19 '13 at 8:04