О расстоянии по вариации между образами вероятностной меры

Let $X$ be a real random variable having absolutely continuous distribution $P,$ and $f,\;g$ be two measurable functions from $\mathbb R^1$ to $\mathbb R^1.$ We are interested in the estimation of the distance in total variation
$$ \Delta := \|P_{f(X)} - P_{g(X)}\|_{\mathrm var} $$
between distributions of random variables $f(X)$ and $g(X)$ in terms of proximity of $f$ and $g.$