  With a view to proving the conjecture of "dlt extension" related to the
abundance conjecture, a sequence of potential candidates for replacing the
Ohsawa measure in the Ohsawa-Takegoshi $L^2$ extension theorem, called the
"lc-measures", which hopefully could provide the $L^2$ estimate of a
holomorphic extension of any suitable holomorphic section on a subvariety with
singular locus, are introduced in the first half of the paper. Based on the
version of $L^2$ extension theorem proved by Demailly, a proof is provided to
show that the lc-measure can replace the Ohsawa measure in the case where the
classical Ohsawa-Takegoshi $L^2$ extension works, with some improvements on the
assumptions on the metrics involved. The second half of the paper provides a
simplified proof of the result of Demailly-Hacon-Paun on the "plt extension"
with the superfluous assumption "$\operatorname{supp} D \subset
\operatorname{supp}(S+B)$" in their result removed. Most arguments in the proof
are readily adopted to the "dlt extension" once the $L^2$ estimates with
respect to the lc-measures of holomorphic extensions of sections on
subvarieties with singular locus are ready.

Complex Variables;Algebraic Geometry