Eritque Arcus Math Notes

Darboux integral

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Def 27.3

for partition 𝛿⊂[𝑎,𝑏]

  1. Δ𝑗=𝑥𝑗−1−𝑥𝑗
  2. inf𝐼𝑓=inf𝑥∈𝐼𝑓(𝑥) and sup𝐼𝑓=sup𝑥∈𝐼𝑓(𝑥)
  3. ∑𝜎=∑𝑛−1𝑗=0Δ𝑗inf𝐼𝑗𝑓 and ∑𝜎=∑𝑛−1𝑗=0Δ𝑗sup𝐼𝑗𝑓 where 𝐼𝑗=[𝑥𝑗,𝑥{𝑗+𝑖}]

Thm 29.1

Then we can take 𝐼=lim𝑘∑𝜏𝑘𝑓 and 𝜏𝑘={𝑥𝑘𝑗∣0≤𝑗≤2𝑘} where 𝜏𝑘⊂𝜏𝑘+1.
Indeed we can say ∑𝜏𝑘+1≤∑𝜏𝑘 and (∑𝜏𝑘) is bounded below and monotone decreasing, which imply it is convergent.
Remarks:

  1. 𝑥𝑘𝑗 means 𝑗-th element in 𝑥𝑘 partition
  2. 𝑥𝑘 partition means we evenly split it into 2𝑘 elements

Lemma 28.2

For two partition 𝜎⊂𝜏

∑𝜎𝑓≤∑𝜏𝑓≤𝑅𝜏(𝑅,𝜉)≤∑𝜏≤∑𝜎𝑓

where 𝑅 is Riemann integral
Basically, if the largest sub-interval is larger, we will get more inaccurate result.

Lemma 28.3

For two partition 𝜎⊂𝜏 , 𝑀=sup(𝑓(𝑥)−𝑓(𝑦)) which is the worst difference where 𝑎≤𝑥,𝑦≤𝑏, we can say ∑𝜏≤∑𝜎𝑓≤∑𝜏+|𝜏|𝑀(#𝜏−#𝜎) where # is the number of that set.

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