Daily Definitions (From class on 11/16/17)

In this class, the topic of Riemann Integration was introduced by Dr. K. This post will cover the definitions and lemmas that were covered during class. Note that the definitions build on another because we slowly built up to the idea of being Riemann Integrable.

1. A function f is bounded on [a,b] if sup\{f(x): x \in [a,b]\} is finite and inf\{f(x):x \in [a,b]\} is finite.

2. Partition P of [a,b] is a set of numbers \{x_0, x_1, \dots , x_n\} with x_o = a,\ x_{k-1} =b, x_{k-1} <x_k\ \forall \ k \in 1, \dots, n

  • Notation: The set of all possible partitions of [a,b] is denoted \mathscr {P}_{[a,b]}

3. A partition Q is a refinement of a given partition P if Q \subseteq P.

  •  Lemma: Under certain conditions, if Q is a refinement of P, then |q-\int_{a}^{b} f| \leq |p-\int_{a}^{b} f|. q \in Q, p \in P

4. Given f, [a,b], P, m_k, \ and\ M_k, we can construct the lower and upper sums for f over [a,b] using P. Note: m_k := inf\{f(x): x\in [x_{k-1},x_k] and M_k := sup\{f(x): x\in [x_{k-1},x_k]

  • Lower Sum for f over [a,b] using P: L(f,[a,b],P)= \sum_{k=1}^{n} m_k (x_k-x_{k-1})
  • Upper Sum for f over [a,b] using P: U(f,[a,b],P)= \sum_{k=1}^{n} M_k (x_k-x_{k-1})
    • Lemma: For a lower sum built from P, the refinement Q of P, then the lower sum of Q satisfies L(f,[a,b], Q) ≥ L(f,[a,b],P). Similarly, for upper bounds, U(f, [a,b], Q) ≤ U(f, [a,b], P).

5.

  • U(f): U(f,[a,b]) = inf \{U (f, [a,b], P): P \in \mathscr {P}_{[a,b]} \} := f_{upper\ over\ [a,b]}
  • L(f): L(f,[a,b]) = sup \{L (f, [a,b], P): P \in \mathscr {P}_{[a,b]} \} := f_{lower\ over\ [a,b]}

 

6. A function f defined on [a,b] is Riemann integrable if L(f) = U(f).

 

Posted on by Shivani Patel
Categories: Definitions

One Response

  1. Jeremy LeCrone says:
    November 26, 2017 at 12:53 pm

    Thank you for the post, but where have you explored these definitions beyond the basic content presented in the lecture? Please recall that daily definitions should explore features of the definition BEYOND basic definitions themselves.

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