Question about equality of Lebesgue integrals

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The discussion centers on proving the equality of Lebesgue integrals for bounded measurable functions g and h defined on measurable sets A and B, respectively. The condition provided is that for all real numbers c, the measure of the sets where g and h exceed c is equal, specifically, \(\mu\{x \in A | g(x) \geq c\} = \mu\{x \in B | h(x) \geq c\}\). The conclusion drawn is that under these conditions, it follows that \(\int_A g = \int_B h\).

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Please assist me with the following. I've been thinking about it for a while, but don't know where to begin.

Let g be a bounded measurable function on a measurable set A,
and h be bounded measurable functions on a measurable set B.

Suppose that [tex]\forall[/tex] c [tex]\in[/tex] R,

[tex]\mu[/tex]{x [tex]\in[/tex] A | g(x) [tex]\geq[/tex] c} = [tex]\mu[/tex]{x [tex]\in[/tex] B | h(x) [tex]\geq[/tex] c}.

Prove that [tex]\int[/tex]A g = [tex]\int[/tex]B h.
 
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There was a small typo, which I just corrected. The problem should make sense now.
 

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