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In short, what I mean is, how can I mention about one certain thing (not one
thing in amount) in Japanese?

For example "one morning" in the following sentence:

> As Gregor Samsa awoke _**one morning**_ from uneasy dreams he found himself
> transformed in his bed into a gigantic insect.

I come across some usages like " **ある** X". For instance:

  * One morning -> **ある** 朝
  * One day -> **ある** 日

Is this **ある** may be what am I looking for?
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expressions prefixes
How to speak about "one" thing in Japanese?
I'd say "yes" to your final question.

You can definitely say something like ある朝 or ある朝のこと as a general way of saying
"one morning".
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> > As Gregor Samsa awoke **_one morning_** from uneasy dreams he found
> himself transformed in his bed into a gigantic insect.
>
> One morning -> ある朝
>
> Is this ある may be what am I looking for?

I can confirm it is the one, as a native speaker.

The kanji version is 或る, but more often it appears in ひらがな: ある人が…, ある時,
あるところにおじいさんとおばあさんが, etc.

We also sometimes say とある: とある日曜の午後 (with rather a feel of popular
literature).

某｛ぼう｝ is yet another possibility, in the same manner: ex. 某日｛ぼうじつ｝,
某会社社長{ぼうかいしゃしゃちょう}, 某ファーストフード店｛てん｝, etc.
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The other answers give good insight into Japanese, but in the particular
example of Kafka's _Metamorphosis_ I can also confirm that ある朝 is the
preferred translation of "one morning":

> **ある朝** 、グレゴール・ザムザが気がかりな夢から目ざめたとき、自分がベッドの上で一匹の巨大な毒虫に変ってしまっているのに気づいた。
>
> **One morning** , as Gregor Samsa was waking up from anxious dreams, he
> discovered that in bed he had been changed into a monstrous verminous bug.
>
> <http://bilinguis.com/book/metamorphosis/jp/en/c1/>
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