[Verse 1] When functions dance toward infinity's door And sequences climb like stairs without end The question burns through analysis lore: Can limits and integrals safely blend? Three theorems guard this sacred exchange Like sentries watching mass escape or stay Each one reveals how convergence can change The rules of mathematical display [Chorus] MCT when they're climbing up Fatou when the path gets rough DCT when there's a roof above Three pillars hold the structure of Limit-integral exchange Nothing lost and nothing strange When you know which tool to choose Integration never lose [Verse 2] Monotone Convergence takes the lead When functions march in ordered ascent Each step builds on what came before indeed No backward slide, no mass misspent The integral follows function's rise Like shadows tracking mountain peaks Patience rewarded, no surprise The limit gives you what it seeks [Chorus] MCT when they're climbing up Fatou when the path gets rough DCT when there's a roof above Three pillars hold the structure of Limit-integral exchange Nothing lost and nothing strange When you know which tool to choose Integration never lose [Verse 3] But chaos reigns when order breaks Fatou's Lemma sounds the alarm Mass can vanish, concentration shakes Functions scatter, lose their charm Liminf below becomes your bound Some measure always slips away In disorder, loss is found But never gain, that's Fatou's way [Bridge] Then Dominated steps between With ceiling fixed above the fray Control prevents the mass machine From letting precious parts decay Universal bound restores the peace Perfect commutation returns Controlled limits find release The theorem every analyst learns [Chorus] MCT when they're climbing up Fatou when the path gets rough DCT when there's a roof above Three pillars hold the structure of Limit-integral exchange Nothing lost and nothing strange When you know which tool to choose Integration never lose [Outro] From probability's expected face To PDEs and quantum fields These three theorems hold their place The power that analysis wields
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