  In 1976, Lai constructed a nontrivial confidence sequence for the mean $\mu$
of a Gaussian distribution with unknown variance $\sigma^2$. Curiously, he
employed both an improper (right Haar) mixture over $\sigma$ and an improper
(flat) mixture over $\mu$. Here, we elaborate carefully on the details of his
construction, which use generalized nonintegrable martingales and an extended
Ville's inequality. While this does yield a sequential t-test, it does not
yield an "e-process" (due to the nonintegrability of his martingale). In this
paper, we develop two new e-processes and confidence sequences for the same
setting: one is a test martingale in a reduced filtration, while the other is
an e-process in the canonical data filtration. These are respectively obtained
by swapping Lai's flat mixture for a Gaussian mixture, and swapping the right
Haar mixture over $\sigma$ with the maximum likelihood estimate under the null,
as done in universal inference. We also analyze the width of resulting
confidence sequences, which have a curious polynomial dependence on the error
probability $\alpha$ that we prove to be not only unavoidable, but (for
universal inference) even better than the classical fixed-sample t-test.
Numerical experiments are provided along the way to compare and contrast the
various approaches, including some recent suboptimal ones.
