# The Lebesgue criterion for Riemann integrability with no oscillations

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.

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I wrote an answer for the case with values in Banach space here: math.stackexchange.com/a/163409/442 –  GEdgar Nov 25 '12 at 22:46