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- I need a little help with Baby Rudin material regarding the convergence of a sequence of sets please

I need a little help with Baby Rudin material regarding the convergence of a sequence of sets please. I wish to follow up on this thread with a definition of convergence of a sequence of sets from Baby Rudin (Principles of Mathematical Analysis, 3rd ed., Rudin) pgs. 304-305:

(pg. 304)

$$E\subset \bigcup_{n=1}^{\infty}A_n$$

Define

(17) $$\mu^* (E) = \inf \sum_{n=1}^\infty \mu (A_n)$$

the inf being taken over all countable coverings of ##E## by open elementary subsets. ##\mu ^* (E)## is called the

(pg. 305)

(22)$$S(A,B)=(A-B)\cup (B-A)$$

(23)$$d(A,B)=\mu^* (S(A,B))$$

We write ##A_n\rightarrow A## if

$$\lim_{n\rightarrow\infty} d(A_n , A) = 0$$

With this notion of limit of a sequence of sets, I wish to do what was conveyed in the post linked in the second sentence from the top of this post, namely I wish to use the Lebesgue's Dominated Convergence Theorem to establish that for some given ##A_n , A\subset \mathbb{R}^p## such that ##A_n\rightarrow A##

(eqn 1) $$\lim_{n\rightarrow\infty}\iint_{A_n}\cdots \int g(\vec{x})d\vec{x} = \iint_{A}\cdots \int g(\vec{x})d\vec{x}$$

Of particular interest is the case of ##S_n := \left\{ (x_1 , x_2 , \ldots , x_N ) : \sum_{k=1}^{N}x_k^{2n}\leq 1\right\}##, I need help on how to show that ##S_n\rightarrow S## for ##S:=\left\{ (x_1 , x_2 , \ldots , x_N ) | -1\leq x_k \leq 1, k=1,2,\ldots , N\right\}##? It's been 20+ years since I've done analysis so any help you can give on how to prove (eqn 1) or help on how to show that ##S_n\rightarrow S## is appreciated. Thanks!

(pg. 304)

**Definition 11.7**) Let ##\mu## be additive, regular, nonnegative, and finite on the ring of elementary sets. Consider countable coverings of any set ##E\subset \mathbb{R}^p## by open elementary subsets ##A_n## :$$E\subset \bigcup_{n=1}^{\infty}A_n$$

Define

(17) $$\mu^* (E) = \inf \sum_{n=1}^\infty \mu (A_n)$$

the inf being taken over all countable coverings of ##E## by open elementary subsets. ##\mu ^* (E)## is called the

*outer measure of E, corresponding to*##\mu##.(pg. 305)

**Definition 11.9**) For any ##A\subset \mathbb{R}^p, B\subset \mathbb{R}^p##, we define(22)$$S(A,B)=(A-B)\cup (B-A)$$

(23)$$d(A,B)=\mu^* (S(A,B))$$

We write ##A_n\rightarrow A## if

$$\lim_{n\rightarrow\infty} d(A_n , A) = 0$$

With this notion of limit of a sequence of sets, I wish to do what was conveyed in the post linked in the second sentence from the top of this post, namely I wish to use the Lebesgue's Dominated Convergence Theorem to establish that for some given ##A_n , A\subset \mathbb{R}^p## such that ##A_n\rightarrow A##

(eqn 1) $$\lim_{n\rightarrow\infty}\iint_{A_n}\cdots \int g(\vec{x})d\vec{x} = \iint_{A}\cdots \int g(\vec{x})d\vec{x}$$

Of particular interest is the case of ##S_n := \left\{ (x_1 , x_2 , \ldots , x_N ) : \sum_{k=1}^{N}x_k^{2n}\leq 1\right\}##, I need help on how to show that ##S_n\rightarrow S## for ##S:=\left\{ (x_1 , x_2 , \ldots , x_N ) | -1\leq x_k \leq 1, k=1,2,\ldots , N\right\}##? It's been 20+ years since I've done analysis so any help you can give on how to prove (eqn 1) or help on how to show that ##S_n\rightarrow S## is appreciated. Thanks!

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