For the Lebesgue Criterion for Riemann integrability, I am trying to prove it without the use of oscillation.
The Lebesgue Criterion for Riemann integrability states that if f:[a,b] is bounded, then f is Riemann integrable iff the set of discontinuities of f has measure 0.
So far I have this:
Let D = {x ∈ [a,b] ǀ f is not continuous on [a,b] at x} Define A(ε, n) = union of intervals in a partition of 2^(n) equal subintervals where $max_{i}$(f) - $min_{i}$(f) with respect to the ith interval is greater than ε
Note D <=> ∀ε > 0, ∃N ∈ $\mathbb{N}$ s.t ∀n ≥ $\mathbb{N}$, x ∈ A(ε,n).
Therefore we can say that D = $\bigcup_{}^{}\\\varepsilon>0$($\bigcap_{}^{}\\n$ (A(ε, n)) = $\bigcup_{}^{}\\k$( $\bigcap_{}^{}\\n$ (A($\frac{1}{k}$, n))
From there I am lost. Any help will be greatly appreciated.
