Pome5E68200DBADDProposition 15.35.19.5E68200DBADD Suppose $\Lambda _ h(f)\ne \varnothing $, put $a_0=\inf \Lambda _ h(f)$, and let $L_ a$ be the left limit of proposition 15.35.18. The set
is at most countable. At every $a>a_0$ with $a\notin D$, the greatest solutions are continuous in $L^1$ and in capacity as $\lambda \to a$, and their equation measures converge in total variation to $\mathrm e^{aU_ a}f\, \mathrm dV$.
Proof. Fix $a_0<\alpha <\beta $, choose an admitted $0<\tau <\alpha $, and use the corrected family $V_\lambda $ from the preceding proof on $[\alpha ,\beta ]$. Its integrals
form a finite bounded increasing real function. For $a\in (\alpha ,\beta ]$, the left-limit proposition gives
If $L_ a\ne U_ a$, this quantity is positive: the integrable nonnegative difference cannot vanish almost everywhere without equality of the canonical potentials. For any finite ordered collection of such points, the sum of their jumps is at most $I(\beta )-I(\alpha )$. Consequently there are only finitely many jumps at least $1/m$ for each positive integer $m$, and at most countably many positive jumps. A countable cover by intervals with rational endpoints inside $(a_0,+\infty )$ proves the assertion about $D$.
If $a\notin D$, the left limit is $U_ a$, and proposition 15.35.17 supplies the same right limit in both topologies and for the equation measures. Splitting an arbitrary approaching sequence into its two sides proves the stated two-sided convergence. $\square$
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