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Rothberger's question of whether the two cardinals $\mathfrak p$ and$\mathfrak t$ are equal, posed back in 1948, was only answered fairly recentlyin the affirmative. Here we answer the more difficult progenitor question(posed in the same place) on when filter-bases can be refined to towers. Ofequal import, our proof that $\mathfrak p=\mathfrak t$ addresses issues raisedby Gowers concerning the "proof path" of the original solution. This is appliedto obtain a gap spectrum result for the ultrapowers $\mathbb N^{\mathbbN}\,/\,{\mathcal U}$.

A Measure Theoretic Proof of $\mathfrak p=\mathfrak t$