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#convergence

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Time lapse video showing the approach of a back door cold front over the Sacramento mountains of southern New Mexico.

This is a classic example of convergence that glider pilots love. Two different temperature air masses are slowly colliding.

This cold front is not very deep, so it behaves like a blunt wedge/spatula as it pushes west. (The high altitude winds are blowing in the opposite direction.)

DOMINATED CONVERGENCE THEOREM
Lebesgue's dominated convergence theorem provides sufficient conditions under which pointwise convergence of a sequence of functions implies convergence of the integrals. It's one of the reasons that makes #Lebesgue integration more powerful than #Riemann integration. The theorem an be stated as follows:

Let \((f_n)\) be a sequence of measurable functions on a measure space \((\mathcal{S},\Sigma,\mu)\). Suppose that \((f_n)\) converges pointwise to a function \(f\) and is dominated by some Lebesgue integrable function \(g\), i.e. \(|f_n(x)|\leq g(x)\ \forall n\) and \(\forall x\in\mathcal{S}\). Then, \(f\) is Lebesgue integrable, and

\[\displaystyle\lim_{n\to\infty}\int_\mathcal{S}f_n\ \mathrm{d}\mu=\int_\mathcal{S}f\ \mathrm{d}\mu\]
#ConvergenceTheorem #Convergence #DominatedConvergenceTheorem #Lebesgue #MeasurableFunction #LebesgueFunction #LebesgueIntegration #RiemannIntegration #MeasureSpace

The Group Works #PatternOfTheDay is Divergence and Convergence Rhythm:

'Diverging widens perspective, explores new terrain and opens up options. Converging coalesces collective wisdom in moving toward focused decisions, concrete outcomes, and the end of the session. Good group process naturally cycles between these two, so be thoughtful about which to engage when.' groupworksdeck.org/patterns/Di #GroupWorksDeck #divergence #convergence

Image ©20?? Randen Pedersen, more info: groupworksdeck.org/train-track