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Measure Theory · Calculus

The Lebesgue Integral

The Lebesgue integral integrates a far broader class of functions than Riemann. Its real power is its behavior under limits — the convergence theorems.

WHAT STUDENTS WILL LEARN
✓Define ∫Xs dμ=∑i=1nai μ(Ai)\int_X s\,d\mu=\sum_{i=1}^{n} a_i\,\mu(A_i) for a simple function s=∑iai1Ais=\sum_i a_i\mathbf{1}_{A_i}
✓Integrate f≥0f\ge 0 as the supremum ∫Xf dμ=sup⁡{∫Xs dμ:0≤s≤f}\int_X f\,d\mu=\sup\{\int_X s\,d\mu:0\le s\le f\}
✓Extend to general ff via f=f+−f−f=f^{+}-f^{-} and ∫f dμ=∫f+dμ−∫f−dμ\int f\,d\mu=\int f^{+}d\mu-\int f^{-}d\mu
✓Compare with Riemann: swap limits and integrals using lim⁡n∫fn dμ=∫f dμ\lim_n\int f_n\,d\mu=\int f\,d\mu
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