* European shares open steady, consolidate recent gains
* Euro money market rates hit lowest on record
* Dollar index holds steady after gaining on higher yields
* Carry trades keep Aussie near 6-mth high vs euro
LONDON, June 10 (Reuters) - An all-time low for euro zone money market rates bolstered the region's bond rally and held down the euro on Tuesday, providing clear evidence that the European Central Bank's latest support measures are gaining traction.
The steady drip-feed of global stimulus also kept world shares inching towards an all-time high as another record close for Wall Street and a three-year high for Asia left them heading for a fifth day of back-to-back gains.
European stocks were happy to take a breather in early trading after gaining almost 10 percent in the last few months, leaving the momentum from last week's ECB cut in interest rates to continue elsewhere.
The rate banks in the euro zone charge one another to borrow overnight - known as EONIA - hit an all-time low at a just-above-zero 0.053 percent, in a move the ECB hopes will feed through to firms and consumers and boost growth.
The euro was pinned near a four-month low against the dollar at $1.3596, while there was a new all-time low for Portuguese bond yields, a proxy of its borrowing costs, just three years after it needed an EU/IMF bailout.
"Broadly what the ECB has done has been pro-risk," said Alvin Tan, a currency strategist at Societe Generale in London.
"Quite apart from the currency moves we have seen, volatility is just plunging and that is all part of the story."
The global appetite for riskier assets has also been whetted by last week's upbeat U.S. non-farm payrolls jobs report.
On Wall Street overnight the S&P 500 ended at a fourth straight record closing high and the Dow at its third.
Aside from the ECB's recent bold moves, there was other reassuring news from the euro zone on Tuesday too.
Italian industrial output rebounded more than expected in April, though France's recovery, which is lagging that of its euro zone peers, only marginally improved in the second quarter according to its central bank.
In Asia, there was muted market reaction to Chinese inflation data as it remained well within the government's comfort zone, giving room for the government to launch fresh stimulus measures if needed to support the economy.
China's consumer prices rose 2.5 percent in May from a year earlier while producer prices fell 1.4 percent.
"No surprises again from May inflation data. Producer prices stabilized ... pointing to muted inflationary pressure," said Andy Ji, senior Asian currency strategist at Commonwealth Bank of Australia in Singapore.
MSCI's broadest index of Asia-Pacific shares outside Japan was up 0.6 percent as trading in the region wound down, leaving it at its highest since June 2011.
Australian shares rose 0.1 percent, but Tokyo's Nikkei and India main stock market bucked the trend as investors cashed in on some of the sizeable gains both have seen recently.
The dollar continued to benefit from rising U.S. Treasury yields. The dollar index, which measures the greenback's strength against a basket of key currencies, held steady after rising 0.2 percent on Monday.
On the day, the dollar stood slightly lower at 102.25 yen . The euro was also flat at $1.3587, while the higher yielding Australian dollar hovered near a six-month high against the euro as hungry investors piled into carry trades.
In commodities, copper steadied after worries about a Chinese probe into metals financing pushed prices to one-month lows in the previous session.
Three-month copper on the London Metal Exchange rose 0.1 percent to $6,680 a tonne from the previous session when it dropped to $6,636 a tonne, its weakest since early May.
Brent crude gained 2 cents to $110.01 a barrel, building on the previous session's sharp 1.3 percent rise made on strong U.S. jobs and improved Chinese export data.
(Reporting by Marc Jones; Editing by Gareth Jones)