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With a view to proving the conjecture of "dlt extension" related to theabundance conjecture, a sequence of potential candidates for replacing theOhsawa measure in the Ohsawa-Takegoshi $L^2$ extension theorem, called the"lc-measures", which hopefully could provide the $L^2$ estimate of aholomorphic extension of any suitable holomorphic section on a subvariety withsingular locus, are introduced in the first half of the paper. Based on theversion of $L^2$ extension theorem proved by Demailly, a proof is provided toshow that the lc-measure can replace the Ohsawa measure in the case where theclassical Ohsawa-Takegoshi $L^2$ extension works, with some improvements on theassumptions on the metrics involved. The second half of the paper provides asimplified 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 proofare readily adopted to the "dlt extension" once the $L^2$ estimates withrespect to the lc-measures of holomorphic extensions of sections onsubvarieties with singular locus are ready.

Extension with log-canonical measures and an improvement to the plt extension of Demailly--Hacon--Paun