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In 1976, Lai constructed a nontrivial confidence sequence for the mean $\mu$of a Gaussian distribution with unknown variance $\sigma$. Curiously, heemployed both an improper (right Haar) mixture over $\sigma$ and an improper(flat) mixture over $\mu$. Here, we elaborate carefully on the details of hisconstruction, which use generalized nonintegrable martingales and an extendedVille's inequality. While this does yield a sequential t-test, it does notyield an ``e-process'' (due to the nonintegrability of his martingale). In thispaper, we develop two new e-processes and confidence sequences for the samesetting: one is a test martingale in a reduced filtration, while the other isan e-process in the canonical data filtration. These are respectively obtainedby swapping Lai's flat mixture for a Gaussian mixture, and swapping the rightHaar mixture over $\sigma$ with the maximum likelihood estimate under the null,as done in universal inference. We also analyze the width of resultingconfidence sequences, which have a curious dependence on the error probability$\alpha$. Numerical experiments are provided along the way to compare andcontrast the various approaches.

Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance