In 上を向いて歩こう Kyu Sakamoto says しあわせは, why doesn't he says しあわせさは? How do you know which 形容動詞 can be written as nouns without さ？
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Although the suffix -さ does turn adjectives into nouns, it also changes the meaning:
Since the suffix changes the meaning, 幸せさ isn't interchangeable with 幸せ. Presumably, the songwriter wrote しあわせ rather than しあわせさ because it had the meaning they wanted to express.
But meaning aside, how can you tell whether a 形容動詞 can also be a noun?
Generally speaking, if a 形容動詞【けいようどうし】 can also be used as a 名詞【めいし】, dictionaries will list them as both. Look for the abbreviations 名 (for 名詞) and 形動 (for 形容動詞), which are used by most monolingual dictionaries. If you're using a dictionary that marks them another way, it should say so somewhere--for example, EDICT uses the abbreviations "adj-na" and "n".
That means 幸せ can also be a 名詞.
You can also look at actual usage, for example by searching a corpus such as the Balanced Corpus of Contemporary Written Japanese (BCCWJ). When I search for 幸せ, I find 5384 results, some of which appear to be using it as a noun. You can use this tool yourself to try to answer this question, if you're still uncertain after consulting a dictionary.