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One can easily make a counterexample, now, by the definition: let $f$ be the characteristic function of the least non-measurable set of reals in the constructible universe $L$, using the canonical definable well-ordering of $L$.

One can easily make a counterexample, now, by the definition: let $f$ be the characteristic function of the least non-measurable set of reals in the constructible universe $L$, using the canonical definable well-ordering of $L$.

Let me point out that this kind of consequence of large cardinals is often pointed to by large cardinal set theorists as evidence that the large cardinal axioms themselves are on the right track, since they provide a such a rich, coherent and desirable structure theory for our everyday mathematics. We infinitely prefer the smooth and elegant descriptive set theory of large cardinals to the awkward land of counterexamples provided by the axiom of constructibility $V=L$.

In particular, assuming that there are sufficient large cardinals, then every projective set of reals is Lebesgue measurable, and this may provide a soft sufficient criterion. The projective statements are those that can be expressed using quantifiers only over the reals and the integers, with the usual algebraic and order structure. Alternatively, the projective sets are those that you get by closing the Borel sets under continuous images and complements.

Let me point out that this kind of consequence of large cardinals is often pointed to by large cardinal set theorists as evidence that the large cardinal axioms themselves are on the right track, since they provide a such a rich, coherent and desirable structure theory for our everyday mathematics. We infinitely prefer the smooth and elegant descriptive set theory of large cardinals to the awkward land of counterexamples provided by the axiom of constructibility $V=L$.

Lastly, let me explain the sense in which your bold statement is on the right track. One of the truly surprising and remarkable discoveries of large cardinal set theory is that the existence of large cardinals has effects on fundamental mathematical truth at the level of descriptive set theory. In particular, the existence of sufficient large cardinals implies that every projectively definable set of reals is Lebesgue measurable. If there is a supercompact cardinal, and much less suffices, as explained in the article Saharon Shelah, Hugh Woodin, Large Cardinals Imply That Every Reasonably Definable Set of Reals Is Lebesgue Measurable, The Bulletin of Symbolic Logic Vol. 8, No. 4 (Dec., 2002), pp. 543-545 (linked to by Andres in the comments), then every set of reals in $L(\mathbb{R})$ is Lebesgue measurable.

Lastly, let me explain the sense in which your bold statement is on the right track. One of the truly surprising and remarkable discoveries of large cardinal set theory is that the existence of large cardinals has effects on fundamental mathematical truth at the level of descriptive set theory. In particular, the existence of sufficient large cardinals implies that every projectively definable set of reals is Lebesgue measurable. If there is a supercompact cardinal, and much less suffices, as explained in the article Saharon Shelah, Hugh Woodin, Large Cardinals Imply That Every Reasonably Definable Set of Reals Is Lebesgue Measurable, Isreal Journal of Mathematics, vol. 70, (1990) pp. 381-394 (reviewed by J. Bagaria in BSL 8:4(2002) pp. 543-545, as linked to by Andres in the comments), then every set of reals in $L(\mathbb{R})$ is Lebesgue measurable. The universe $L(\mathbb{R})$ consists of those sets that are constructible relative to reals.