The auxiliary verb (助動詞) ～けむ is probably a combination of the mizenkei of the perfect aspect marker ～き and dubitative auxiliary verb ～む. As such, it is governed by the ren’youkei. Though it is often grouped with ～らむ, it lacks the evidentiality that ～らむ has. It conjugates as a defective yodan verb lacking the mizenkei, ren’youkei, and meireikei.
The conjunctive particle ～ば cannot attach to the izenkei. ～けむ is also not to be confused with the old mizekei of adjectives (～け) and dubitative ～む (which would subsequently mean “is probably <adj>”). ～けむ has two uses:
- 過去疑義 Past Dubitative Mood
This is very simply expressing doubt or stating very general hearsay (this frequently being the rentaikei) about a past event. The translation would be ～ただろう、～たとかいう、～たようだ.
いにしへにありけむ人も （遠い昔にいたという人も) The people who are said to have lived in the distant past also…
This use may appear paraphrastically to indirectly state a past that is actually known.
- 過去の事情の推量 Past Inferential Mood of Cirumstance
Often (but not always) appearing with an interrogative word or adverb, this use expresses a conjecture or reflection about a past event with reference to cause, reason, degree, time, place, method, or person(s) involved. The resulting sentence can be interrogative (i.e. How nice must it have been?) or declarative (i.e. How nice it must have been!) in nature. It is more common that the above usage.
これをいみじと思へばこそ、記しとどめて世にも伝えけめ (これを素晴らしいと思ったからこそ、本に書き記して後世にも伝えたのだろう。) I wonder if they wrote it down and conveyed it to their posterity for the very reason that they thought it was wonderful.) [reason – explicit]
妻別れ悲しくありけむ (妻と別れてなんて悲しかっただろう。) How sad it must have been to be separated from his wife! [degree-unexpressed]
君や来し我や行きけむ (あなたが来たか、私が行ったか。) Did you come to me, or did I go to you I wonder? [people involved – explicit]
Classical Japanese 