0.2. Criteria for Riemann Integrability 5
that any function that is Riemann integrable remains integrable and
the value of the integral does not change.
0.2. Criteria for Riemann Integrability
These conditions are typically discussed in texts used for a first course
in real analysis. They are presented here to give us a framework on
how to proceed once we encounter Lebesgue integration. More details
can be found in Abbott [1]orLay[9].
One of our first goals is to show that the lower integral is always
less than or equal to the upper integral. (This may seem obvious
to the reader, but one must always be able to provide a proof of the
obvious. After all, mathematics occasionally has surprises in store for
us!) In order to establish this claim, we will be comparing lower sums
and upper sums. Although we know from Proposition 0.1.6, given a
partition, that the lower sum is always less than or equal to the upper
sum, we have not yet compared the lower sum for one partition with
the upper sum for a possibly different partition. To do this, we need
the notion of a refinement.
Definition 0.2.1. The partition P
∗
of [a, b]isarefinement of the
partition P if each point in P is also in P
∗
. P
∗
is a common re-
finement of the partitions P
1
and P
2
if P
∗
is a refinement of each of
P
1
and P
2
.
Given any two partitions P
1
and P
2
of [a, b] we can take the points
in P
1
∪P
2
, put them in increasing order, and form a new partition of
[a, b]. This new partition will then be a common refinement of P
1
and
P
2
. Hence, given any two partitions, there always exists a common
refinement.
Lemma 0.2.2. Let f ∈ B[a, b].
i) If P
∗
is a refinement of the partition P of [a, b],then
L(f,P) ≤ L(f,P
∗
) ≤ U(f, P
∗
) ≤ U(f, P) .
ii) If P
1
and P
2
are any two partitions of [a, b],then
L(f,P
1
) ≤ U(f, P
2
) .
剩余232页未读，继续阅读
sudi2006435
- 粉丝: 0
- 资源: 3
我的内容管理
展开
最新资源
- 中国微型数字传声器：技术革新与市场前景
- 智能安防：基于Hi3515的嵌入式云台控制系统设计
- 手机电量低时辐射真增千倍？解析手机使用谣言
- 56F803型DSP驱动的高精度大功率超声波电源控制策略研究
- ARM与GPRS结合的远程监测系统设计
- GPS与RFID技术结合的智能巡检系统设计
- CPLD驱动的低功耗爆炸场温度测试系统设计
- 基于FPGA的智能驱动控制系统：可扩展设计与工业网络协议
- 基于ATmega128和CH374的嵌入式USB接口设计
- 基于AT89C52的温度补偿超声波测距仪：高精度设计与应用
- MSP430F448单片机在交流数字电压表中的应用
- 提升变频器应用效率的12项实用技巧
- STM32F103在数字电镀电源并联均流系统中的应用
- PSpice仿真下的升压开关电源设计：拓扑分析与CCM稳定性提升
- 轻巧高效：MSP430主导的低成本无线传感器网络节点设计
- FPGA在EDA/PLD中实现LVDS接口的应用解析
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
