# dealing infinities in the book frank jones lebesgue integration on euclidean spaces

I am currently reading the book "Frank Jones : Lebesgue Integeartion on Euclidean Spaces". The writing is not completely rigorous. for example measure can be $$\infty$$ but he doesn't tell what to do in such cases.

what does the following statement mean when one of $$\lambda(A_i) = \infty.$$ (I mean how does infinite series defined when one of them is $$\infty$$)

$$\lambda\Big( \bigcup_{i=1}^{\infty} A_i \Big) = \sum_{i=1}^{\infty} \lambda(A_i).$$

It is particularly confusing when argument depends on measure of set not being $$\infty$$

let $$A_1 \subset A_2 \subset A_3 \dots$$ then $$\displaystyle \lambda \Big( \bigcup_{n=1}^{\infty} A_n \Big) = \lim_{N \rightarrow \infty} \lambda(A_N)$$

we can write union of $$A_i$$ as disjoint union as follows $$\displaystyle\bigcup_{n=1}^{\infty} A_n = A_{1} \cup \bigcup_{n=1}^{\infty} (A_{n+1} \sim A_{n})$$ then we have that

$$\displaystyle \lambda(\bigcup_{n=1}^{\infty} A_i) = \lambda(A_{1}) + \sum_{n=1}^{\infty} \lambda(A_{n+1} \sim A_{n})$$ he then says that

$$= \displaystyle \lim_{N \rightarrow \infty} \Big[ \lambda(A_{1}) + \sum_{n=1}^{N} \lambda(A_{n+1} \sim A_{n}) \Big]$$

$$= \displaystyle \lim_{N \rightarrow \infty} \lambda \Big( A_{1} \cup \bigcup_{n=1}^{N} (A_{n+1} \sim A_{n}) \Big)$$

$$= \displaystyle \lim_{N \rightarrow \infty} \lambda(A_N)$$

then following is the excercise

Let $$A_1 \supset A_2 \supset A_3 \dots$$ if $$\lambda(A_1) < \infty$$ then we have that $$\displaystyle \lambda \Big( \bigcap_{n=1}^{\infty} A_n \Big) = \lim_{n \rightarrow \infty} \lambda(A_i))$$

now we know that $$\displaystyle A_{K} = \Big( \bigcap_{n=1}^{\infty} A_n \Big) \cup \Big(\bigcup_{n=k}^{\infty}(A_{n} \sim A_{n+1}) \Big)$$

so we have that $$\lambda(A_{K}) = \lambda(A) + \sum_{n=k}^{\infty} \lambda(A_{n} \sim A_{n+1})$$

and $$\displaystyle \lim_{K \rightarrow \infty} A_{K} = \lim_{K \rightarrow \infty} \Big[ \lambda(A) + \sum_{n=k}^{\infty} \lambda(A_{n} \sim A_{n+1}) \Big] = \lambda(A)$$

But I didn't use the condition that $$\lambda(A_1) < \infty.$$ ?? what is wrong with the arugment. and I am really confused handling with limit of sequence of numbers which contains $$\infty.$$ Book doesn't talk about what it means for limit when sequence contains $$\infty$$ or handle it separately.

• "what does the following statement mean when one of $\lambda(A_i)=\infty$?" It means $\infty=\infty$. Commented Jul 29, 2020 at 13:02
• what is limit of following sequence $(0,\infty,0,0,0,0,\dots )$ is it $0$?? Commented Jul 29, 2020 at 13:05
• See page 25: "the measure of $A$ will be a nonnegative real number or $\infty$." And see page 29-30. Commented Jul 29, 2020 at 13:07
• If you are in page 44-on, you are working with sets having finite outer measure, i.e. such that $\lambda^*(A) < \infty$ Commented Jul 29, 2020 at 13:16
• Presumably, as is explicitly said in some other text, if we have an unbounded family of nonnegative numbers $\{ a_i \}$ we have to set $\Sigma a_i = \infty$. Commented Jul 29, 2020 at 13:26

For your last example, the assumption that $$\lambda(A_j)<\infty$$ for some $$j$$ is crucial.
Let $$\lambda$$ be Lebesgue measure and $$A_k$$ be the interval $$(k,\infty)$$. Then each $$\lambda(A_k)=\infty$$ but $$\bigcap_k A_k=\emptyset$$. The usual error here is to assume that "$$\infty-\infty$$" equals $$0$$.