  The classical Liouvile integrability means that there exist $n$ independent
first integrals in involution for $2n$-dimensional phase space. However, in the
infinite-dimensional case, an infinite number of independent first integrals in
involution don't indicate that the system is solvable. How many first integrals
do we need in order to make the system solvable? To answer the question, we
obtain an infinite dimensional Hamilton-Jacobi theory, and prove an infinite
dimensional Liouville theorem. Based on the theorem, we give a modified
definition of the Liouville integrability in infinite dimension. We call it the
$L$-integrability. As examples, we prove that the string vibration equation and
the KdV equation are $L$-integrable. In general, we show that an infinite
number of integrals is complete if all action variables of a Hamilton system
can reconstructed by the set of first integrals.
