[Verse 1] Picture a sphere spinning in space tonight Rotate any region, the area stays right Translation on the real line, length preserved These aren't coincidences, there's a pattern observed [Pre-Chorus] Every group that's locally compact and clean Has a measure waiting to be seen [Chorus] Haar measure lives where symmetry flows Left invariant, that's how it goes Rotate or translate, the volume's the same Symmetry determines the measuring game Haar measure, Haar measure Making groups complete Haar measure, Haar measure Where algebra and analysis meet [Verse 2] Start with compact sets, approximate from outside Build finitely additive, take it in stride Compactness argument or ultrafilter's might Produces the measure that gets it just right [Pre-Chorus] Countably additive, that's the goal Invariance makes the structure whole [Chorus] Haar measure lives where symmetry flows Left invariant, that's how it goes Rotate or translate, the volume's the same Symmetry determines the measuring game Haar measure, Haar measure Making groups complete Haar measure, Haar measure Where algebra and analysis meet [Bridge] Uniqueness follows from the invariant way Lebesgue measure's just the real line case Harmonic analysis builds upon this base Fourier transforms find their rightful place From representation theory to the Langlands dream Adelic measures in the number theory scheme [Chorus] Haar measure lives where symmetry flows Left invariant, that's how it goes Approximate outside then take the limit Symmetry constrains what lives within it Haar measure, Haar measure Foundation so deep Haar measure, Haar measure Makes calculus on groups complete [Outro] When symmetry acts upon a space There's one natural way to measure its grace Haar's theorem proves what we always knew Symmetry tells us how to measure true
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