## 0.1 “Limit”——footstone of mathematical analysis

• In Group Theory’’, the basic operation is $$"\times"$$, we define the kind of mapping which keeps the operation as : homomorphic mapping. It is a key mapping in Group Theory.

• In Mathematical Analysis, the foodstone is limit and the continuous functions are that keep limit operation, i.e.

$if\ x_{n}\longrightarrow x\ \ \Rightarrow f(x_{n})\longrightarrow f(x)$

So we can see the importance of continuous fuctions in Mathematical Analysis.

• Different definition of Continuous Functions:

Assume there are two metric space $$(E,\ d)$$ , and $$(S,\ \rho)$$.

$$f:\ E\longrightarrow S$$ then the following proposition are equivalent:

1. The definition of continuous functtion that most of us learnt in Mathematical Analysis:

$\forall\epsilon>0,\ \exists\delta>0,\ such\ that\ f(B_{d}(x_{0},\delta))\subset B_{\rho}(f(x_{0}),\epsilon)$

Remark: Most of us are used to the definition in $$(R,\ d)$$. However, it is only a case of metric space. Though it helps to understand by realine $$R,$$ or $$R^{2}$$, $$R^{3}$$……, metric space is much broader than that.

1. $$\forall G\subset S$$, $$G$$ is open, we have $$f^{-1}(G)=\{x\in E:\ f(x)\in G\}$$ is open subset of $$E$$
2. $$\forall F\subset S$$, $$F$$ is closed, we have $$f^{-1}(F)=\{x\in E:\ f(x)\in F\}$$ is closed subset of $$E$$

## 0.2 Riemann integral and Lebesgue integral

We learnt in Mathematical Analysis about Riemann integral. In Measure Theory, there is another integral called Lebesgue integral. It is natural to ask : Why is Lebesgue integral needed?

1. Riemann integral is partitioned by the range of function f; Lebesgue integral is prtitioned by the domain of function f

It is like two persons are counting their money. What Mr. Riemann does is: add the cashes one by one in order. What Mr. Lebesgue does is: sort the cash, put cashes with the same value together then count the number of $100, number of$50…….. What Lebesgue does seems to be more effective.

1. Riemann integrable function space is not complete but the sapce for Lebesgue integrable function is complete. i.e, the limitation of a convergence sequence of Riemann integrable functions may not be Riemann integrable. But the limitation of a convergence sequence of Lebesgue integrable is Lebesgue integrable.

2. Riemann integral requires stronger precondition to interchange the order of limitaion and integration that is: uniform convergence. In Lebesgue, we only need Lebesgues dominated convergence theorem (DCT)’’

3. Let f be a bounded function on a bounded interval [a, b]. Then

• f is Riemann integrable on [a,b] iff f is continuous i.e (m) on [a, b]

You need to show that: upper-Riemann integral=lower-Riemann integral

Proof (to be cont)

• f is Lebesgue integrable on [a, b] and the Lebesgue integral $$\int_{[a,b]}fdm$$ equals the Riemann integral $$\oint_{[a,b]}f$$, i.e., the two integrals coincide.
1. If f is Riemann absolute integrabe on an unbounded area $$G$$ and on every bounded subarea of $$G$$ is Riemann integrable, then f is Lebesgue integrable on $$G$$ and the two integrals are coincide.

To show this, you need to use MCT and DCT

For above $$f(x_{1},x_{2},...,x_{n})$$, Riemann absolute integrable is equivalent to Riemann intnergable when $$n\geq2$$}

## 0.3 Examples

$f(x)=\begin{cases} \begin{array}{cc} 1 & x\in Q\\ 0 & x\in R/Q \end{array}\end{cases}$

$$f(x)$$ Legesbue integrable but not Riemann integrable}

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